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Poisson summation formula

The Poisson summation formula is a fundamental identity in and that relates the discrete sum of a over the integers to the discrete sum of its over the integers. For a f (rapidly decaying and smooth), it asserts that \sum_{n=-\infty}^{\infty} f(n) = \sum_{n=-\infty}^{\infty} \hat{f}(n), where \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx denotes the of f. This equivalence holds under suitable decay conditions on f to ensure , and it can be derived from the periodicity of the summed and properties of the . Named after the French mathematician , the formula first appeared in his 1823 memoir "Continuation of the essay on definite integrals and the summation of series," where it arose in evaluating remainders for Euler's summation formula. Although Poisson provided an early form involving integrals and trigonometric series for functions of , the modern version in terms of Fourier transforms was formalized in the , particularly through the work on Schwartz distributions and . The result generalizes to higher dimensions and more abstract settings, such as locally compact abelian groups, where it connects sums over lattice points to integrals over dual groups. The formula's significance lies in its bridging of discrete and continuous mathematics, enabling powerful applications across fields. In , it underpins the of the via the , improving error bounds in problems like the divisor sum and circle problem. It also appears in the study of L-functions, where it helps analyze the distribution of values following , and in theory for characteristic polynomials. Beyond , generalizations find use in for sampling theorems, error-correcting codes, and . Voronoi's extensions in the early incorporated weights and , further broadening its scope in summation problems.

Core Formulation

Basic Statement

The Poisson summation formula establishes a deep connection in Fourier analysis between the discrete sum of a function evaluated at integer lattice points and the sum of its Fourier transform at those same points. For a Schwartz function f: \mathbb{R}^n \to \mathbb{C}, which consists of infinitely differentiable functions together with all their derivatives decaying faster than any polynomial at infinity, the formula asserts \sum_{m \in \mathbb{Z}^n} f(m) = \sum_{k \in \mathbb{Z}^n} \hat{f}(k), where the Fourier transform \hat{f} is given by the convention \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) \, e^{-2\pi i \, x \cdot \xi} \, dx. This relation serves as a bridge between discrete lattice sums, which arise in number theory and sampling theory, and continuous integrals via the Fourier transform, underscoring the periodic structure inherent in the integer lattice and enabling transformations between time and frequency domains in higher dimensions. A classic illustration occurs with the Gaussian function f(x) = e^{-\pi \|x\|^2} on \mathbb{R}^n, which is self-dual under the Fourier transform, satisfying \hat{f}(\xi) = e^{-\pi \|\xi\|^2}. Applying the formula then yields \sum_{m \in \mathbb{Z}^n} e^{-\pi \|m\|^2} = \sum_{k \in \mathbb{Z}^n} e^{-\pi \|k\|^2}, a tautological equality that nonetheless confirms the formula's consistency and highlights its role in evaluating theta functions in one dimension.

Periodization Approach

One approach to deriving the Poisson summation formula involves periodizing a function defined on the real line \mathbb{R}, transforming it into a suitable for expansion. For a f: \mathbb{R} \to \mathbb{C} that is integrable (i.e., f \in L^1(\mathbb{R})), the periodized version g: \mathbb{R} \to \mathbb{C} is defined by summing over translates: g(x) = \sum_{m \in \mathbb{Z}} f(x + m). This sum converges for almost every x \in [0,1] under the L^1 assumption, as the tails decay due to integrability. Since g is 1-periodic, it admits a expansion on the \mathbb{T} = \mathbb{R}/\mathbb{Z}. The Fourier coefficients c_k of g for k \in \mathbb{Z} are given by c_k = \int_0^1 g(x) e^{-2\pi i k x} \, dx = \hat{f}(k), where \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx denotes the of f (using the standard normalization). Thus, the of g is g(x) = \sum_{k \in \mathbb{Z}} \hat{f}(k) e^{2\pi i k x}, valid pointwise under suitable conditions on f. Equating the two expressions for g(x) yields \sum_{m \in \mathbb{Z}} f(x + m) = \sum_{k \in \mathbb{Z}} \hat{f}(k) e^{2\pi i k x}. Evaluating at x = 0 recovers the basic Poisson summation formula: \sum_{m \in \mathbb{Z}} f(m) = \sum_{k \in \mathbb{Z}} \hat{f}(k). This periodization extends naturally to higher dimensions. For f: \mathbb{R}^n \to \mathbb{C} integrable on \mathbb{R}^n, the periodized function over the \mathbb{Z}^n is g(x) = \sum_{m \in \mathbb{Z}^n} f(x + m), \quad x \in [0,1]^n, which is periodic with respect to the unit \mathbb{T}^n. The Fourier coefficients are then c_k = \hat{f}(k) for k \in \mathbb{Z}^n, leading to \sum_{m \in \mathbb{Z}^n} f(x + m) = \sum_{k \in \mathbb{Z}^n} \hat{f}(k) e^{2\pi i k \cdot x}, and at x = 0, \sum_{m \in \mathbb{Z}^n} f(m) = \sum_{k \in \mathbb{Z}^n} \hat{f}(k). The multidimensional is \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i x \cdot \xi} \, dx. For the interchange of sums in the periodization and Fourier series to be justified by absolute convergence, f is typically taken from the \mathcal{S}(\mathbb{R}^n) of smooth functions with rapid decay (i.e., \sup_{x \in \mathbb{R}^n} |x^\alpha \partial^\beta f(x)| < \infty for all multi-indices \alpha, \beta). In this case, both \sum_{m \in \mathbb{Z}^n} |f(m)| and \sum_{k \in \mathbb{Z}^n} |\hat{f}(k)| converge absolutely, ensuring the equality holds without conditional convergence issues.

Derivations

Fourier Series Derivation

The Poisson summation formula can be derived by considering the Fourier series expansion of a periodized version of a rapidly decreasing function. Assume f belongs to the Schwartz space \mathcal{S}(\mathbb{R}), consisting of smooth functions that decay faster than any polynomial along with all their derivatives. Define the periodic function g: \mathbb{R} \to \mathbb{C} with period 1 by g(x) = \sum_{m=-\infty}^{\infty} f(x + m), \quad x \in [0,1). $&#36; Due to the rapid decay of $f$, the series defining $g$ converges uniformly on $\mathbb{R}$, ensuring $g$ is continuous and bounded.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf)[](http://kryakin.site/am2/Stein-Shakarchi-1-Fourier_Analysis.pdf) The Fourier series of $g$ on $[0,1)$ is given by g(x) = \sum_{k=-\infty}^{\infty} c_k e^{2\pi i k x}, where the coefficients are c_k = \int_0^1 g(x) e^{-2\pi i k x} , dx. Substituting the expression for $g$, we obtain c_k = \int_0^1 \left( \sum_{m=-\infty}^{\infty} f(x + m) \right) e^{-2\pi i k x} , dx. The rapid decay of $f$ justifies interchanging the sum and integral, yielding $&#36; c_k = \sum_{m=-\infty}^{\infty} \int_0^1 f(x + m) e^{-2\pi i k x} \, dx.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf) For each fixed m, substitute y = x + m, so dx = dy and the limits shift from x \in [0,1) to y \in [m, m+1). This gives \int_0^1 f(x + m) e^{-2\pi i k x} \, dx = \int_m^{m+1} f(y) e^{-2\pi i k (y - m)} \, dy = e^{2\pi i k m} \int_m^{m+1} f(y) e^{-2\pi i k y} \, dy. Since k, m \in \mathbb{Z}, e^{2\pi i k m} = 1, so c_k = \sum_{m=-\infty}^{\infty} \int_m^{m+1} f(y) e^{-2\pi i k y} \, dy = \int_{-\infty}^{\infty} f(y) e^{-2\pi i k y} \, dy = \hat{f}(k), $&#36; where $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(y) e^{-2\pi i \xi y} \, dy$ is the Fourier transform of $f$. The interchange of sum and integral over the real line is valid due to the absolute integrability of $f$.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf)[](http://kryakin.site/am2/Stein-Shakarchi-1-Fourier_Analysis.pdf) The rapid decay of $f$ implies that the Fourier coefficients $c_k = \hat{f}(k)$ also decay rapidly, ensuring uniform and absolute convergence of the Fourier series for $g$ on $[0,1)$.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf) Thus, evaluating the series at $x = 0$ yields g(0) = \sum_{k=-\infty}^{\infty} c_k = \sum_{k=-\infty}^{\infty} \hat{f}(k). On the other hand, g(0) = \sum_{m=-\infty}^{\infty} f(0 + m) = \sum_{m=-\infty}^{\infty} f(m), which converges absolutely due to the decay of $f$. Equating the two expressions gives the Poisson summation formula: \sum_{m=-\infty}^{\infty} f(m) = \sum_{k=-\infty}^{\infty} \hat{f}(k). ### Fourier Transform Derivation The Poisson summation formula can be derived using the Fourier transform on the real line, leveraging the self-duality of the Dirac comb distribution. Consider the Fourier transform defined for a Schwartz function $f \in \mathcal{S}(\mathbb{R})$ by \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} , dx, with the inversion theorem stating that f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i x \xi} , d\xi. This convention ensures the transform is unitary on L^2(\mathbb{R}) and extends naturally to tempered distributions. To connect discrete sums to the continuous transform, introduce the Dirac comb (or Shah function) \eta(x) = \sum_{k \in \mathbb{Z}} \delta(x - k), a tempered distribution representing periodic Dirac deltas at integers. Its Fourier transform is [ \hat{\eta}(\xi) = \sum_{m \in \mathbb{Z}} \delta(\xi - m), exhibiting self-duality under this normalization; this follows from the fact that $\eta$ is periodic with period 1, and its Fourier series coefficients yield the dual comb in the frequency domain.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) More precisely, the transform of each $\delta(x - k)$ is $e^{-2\pi i k \xi}$, so formally summing gives a periodic train that equals the dual comb in the distributional sense.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) Now consider the periodized sum $g(x) = \sum_{n \in \mathbb{Z}} f(x + n) = (f * \eta)(x)$, the convolution of $f$ with the comb, which is well-defined and smooth for $f \in \mathcal{S}(\mathbb{R})$ due to rapid decay. Taking the Fourier transform yields \hat{g}(\xi) = \hat{f}(\xi) \hat{\eta}(\xi) = \hat{f}(\xi) \sum_{m \in \mathbb{Z}} \delta(\xi - m), since convolution becomes multiplication in the frequency domain.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) Applying the inversion theorem to $\hat{g}$ gives g(x) = \int_{-\infty}^{\infty} \hat{g}(\xi) e^{2\pi i x \xi} , d\xi = \sum_{m \in \mathbb{Z}} \hat{f}(m) e^{2\pi i m x}, as the integral against the comb sifts $\hat{f}$ at integers.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) Evaluating at $x = 0$ produces g(0) = \sum_{n \in \mathbb{Z}} f(n) = \sum_{m \in \mathbb{Z}} \hat{f}(m), which is the one-dimensional Poisson summation formula. This holds for all Schwartz functions, with absolute convergence of both sides guaranteed by the decay properties of $f$ and $\hat{f}$.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) For the $n$-dimensional case, the derivation generalizes analogously over $\mathbb{R}^n$. The integer lattice comb is $\eta(\mathbf{x}) = \sum_{\mathbf{k} \in \mathbb{Z}^n} \delta(\mathbf{x} - \mathbf{k})$, with Fourier transform $\hat{\eta}(\boldsymbol{\xi}) = \sum_{\mathbf{m} \in \mathbb{Z}^n} \delta(\boldsymbol{\xi} - \mathbf{m})$. For $f \in \mathcal{S}(\mathbb{R}^n)$, the periodized $g(\mathbf{x}) = \sum_{\mathbf{k} \in \mathbb{Z}^n} f(\mathbf{x} + \mathbf{k})$ has transform $\hat{g}(\boldsymbol{\xi}) = \hat{f}(\boldsymbol{\xi}) \hat{\eta}(\boldsymbol{\xi})$, and inversion at $\mathbf{x} = \mathbf{0}$ yields $\sum_{\mathbf{k} \in \mathbb{Z}^n} f(\mathbf{k}) = \sum_{\mathbf{m} \in \mathbb{Z}^n} \hat{f}(\mathbf{m})$.[](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf) ## Advanced Formulations ### Distributional Formulation The distributional formulation of the Poisson summation formula extends the classical result to the framework of tempered distributions, enabling its application to functions that may not satisfy pointwise summability conditions. Central to this approach is the [Dirac comb](/page/Dirac_comb) distribution $D$, defined on the [Schwartz space](/page/Schwartz_space) $\mathcal{S}(\mathbb{R})$ as $D = \sum_{m \in \mathbb{Z}} \delta_m$, where $\delta_m$ denotes the [Dirac delta distribution](/page/Dirac_delta_distribution) centered at $m$. This distribution acts on test functions $\phi \in \mathcal{S}(\mathbb{R})$ via the pairing $\langle D, \phi \rangle = \sum_{m \in \mathbb{Z}} \phi(m)$, representing a formal summation over integer points without requiring absolute convergence of the series.[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf)[](https://arxiv.org/pdf/2506.18638) The Fourier transform of the Dirac comb, denoted $\hat{D}$, satisfies the remarkable self-duality property $\hat{D}(\xi) = \sum_{k \in \mathbb{Z}} \delta_k$, which is itself an instance of the [Poisson summation formula](/page/Poisson_summation_formula) in the distributional sense. This equality holds because the Fourier transform of a sum of translated deltas yields another sum of deltas at integer frequencies, leveraging the periodicity inherent in the lattice $\mathbb{Z}$. For a general tempered distribution $T \in \mathcal{S}'(\mathbb{R})$, the Poisson relation manifests as $\langle T, D \rangle = \langle \hat{T}, \hat{D} \rangle$, where $\hat{T}$ is the Fourier transform of $T$. Specializing to the action on test functions, this implies $\sum_{m \in \mathbb{Z}} \langle T, \delta_m \rangle = \sum_{k \in \mathbb{Z}} \langle \hat{T}, \delta_k \rangle$, or equivalently, $\sum_{m \in \mathbb{Z}} T(m) = \sum_{k \in \mathbb{Z}} \hat{T}(k)$ in the distributional pairing sense.[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf)[](https://arxiv.org/pdf/2506.18638)[](https://bigwww.epfl.ch/publications/nguyen1502.pdf) For functions $f \in \mathcal{S}(\mathbb{R})$, the extension arises by considering the distribution $T_f = \sum_{m \in \mathbb{Z}} f(m) \delta(x - m)$, whose Fourier transform is $\hat{T}_f(\xi) = \sum_{k \in \mathbb{Z}} \hat{f}(k) \delta(\xi - k)$. This distributional equality $\hat{T}_f = T_{\hat{f}}$ encapsulates the [Poisson summation formula](/page/Poisson_summation_formula), holding via convergence in the weak topology of $\mathcal{S}'(\mathbb{R})$ rather than pointwise or absolute summability of the series involved. The key advantage is that it applies broadly to rapidly decreasing smooth functions without additional regularity assumptions beyond membership in the [Schwartz class](/page/Schwartz_class), providing a rigorous foundation for generalizations to [tempered distributions](/page/Tempered_distribution).[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf)[](https://arxiv.org/pdf/2506.18638) ### Formulation for Tempered Distributions Tempered distributions are continuous linear functionals on the [Schwartz space](/page/Schwartz_space) $\mathcal{S}(\mathbb{R})$, equipped with the topology induced by the family of seminorms $\nu_{m,n}(f) = \sup_{x \in \mathbb{R}} (1 + x^2)^m |f^{(n)}(x)|$, where $f^{(n)}$ denotes the $n$-th derivative of $f$. These distributions encompass functions of polynomial growth, including constants, and allow the [Fourier transform](/page/Fourier_transform) to be defined via duality: for a tempered distribution $T \in \mathcal{S}'(\mathbb{R})$, its [Fourier transform](/page/Fourier_transform) $\hat{T}$ is given by $\langle \hat{T}, \phi \rangle = \langle T, \hat{\phi} \rangle$ for all $\phi \in \mathcal{S}(\mathbb{R})$, where $\hat{\phi}(x) = \int_{\mathbb{R}} \phi(y) e^{-2\pi i x y} \, dy$. This extension preserves the structure of $\mathcal{S}'(\mathbb{R})$ as the [Fourier transform](/page/Fourier_transform) is a continuous isomorphism on this space.[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf) The Poisson summation formula in this setting relates a tempered distribution $T$ to the Dirac comb $\sum_{m \in \mathbb{Z}} \delta_m$, where $\delta_m$ is the Dirac delta at $m$. Specifically, the formula states that $\langle T, \sum_{m \in \mathbb{Z}} \delta_m \rangle = \langle \hat{T}, \sum_{k \in \mathbb{Z}} \delta_k \rangle$, provided the pairings are well-defined in the distributional sense. The left side evaluates to $\sum_{m \in \mathbb{Z}} T(m)$, while the right side is $\sum_{k \in \mathbb{Z}} \hat{T}(k)$. The Dirac comb itself is a tempered distribution because for any $\phi \in \mathcal{S}(\mathbb{R})$, the sum $\sum_{m \in \mathbb{Z}} \phi(m)$ converges absolutely due to the rapid decay of $\phi$, and its Fourier transform is again the Dirac comb (up to normalization), ensuring the formula holds under the duality definition. This equality follows from the general distributional formulation but requires the growth controls of tempered distributions to handle slowly growing functions at infinity.[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf) A notable example involves the constant function $T = 1$, which defines a tempered distribution via $\langle 1, \phi \rangle = \int_{\mathbb{R}} \phi(x) \, dx$, as the integral is bounded by the seminorms on $\mathcal{S}(\mathbb{R})$. Its Fourier transform is the Dirac delta at zero: $\hat{1} = \delta_0$. However, the direct pairing $\langle 1, \sum_{m \in \mathbb{Z}} \delta_m \rangle = \sum_{m \in \mathbb{Z}} 1$ diverges, necessitating regularization, such as the principal value sense, to interpret the summation. Applying the formula in this regularized context leads to important identities, including the partial fraction expansion $\pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^\infty \left( \frac{1}{z - n} + \frac{1}{z + n} \right)$ for non-integer $z \in \mathbb{C}$, derived by considering the periodization of suitable functions like $\frac{\pi^2}{\sin^2(\pi (z + x))}$ and integrating the resulting Poisson-applied sum. This expansion arises from the distributional limit and highlights the formula's utility for unbounded functions with polynomial bounds.[](https://www.math.cuhk.edu.hk/course_builder/1617/math4060/Poisson%20summation.pdf) In multiple dimensions, the formulation extends naturally to $\mathbb{R}^n$. Tempered distributions $T \in \mathcal{S}'(\mathbb{R}^n)$ satisfy continuity with respect to the seminorms $\nu_{m,\alpha}(f) = \sup_{x \in \mathbb{R}^n} (1 + |x|^2)^m |D^\alpha f(x)|$, where $\alpha$ is a multi-index. The Poisson formula becomes $\langle T, \sum_{m \in \mathbb{Z}^n} \delta_m \rangle = \langle \hat{T}, \sum_{k \in \mathbb{Z}^n} \delta_k \rangle$, with the Fourier transform defined dually using the $n$-dimensional integral $\hat{\phi}(\xi) = \int_{\mathbb{R}^n} \phi(y) e^{-2\pi i x \cdot \xi} \, dy$. The Dirac comb on $\mathbb{Z}^n$ remains tempered due to rapid decay ensuring convergence of $\sum_{m \in \mathbb{Z}^n} \phi(m)$, and the formula applies to distributions with tempered growth at infinity, such as polynomials, under appropriate regularization for divergent sums.[](https://www-users.cse.umn.edu/~garrett/m/fun/poisson.pdf) ## Applicability Conditions ### Convergence Requirements The Poisson summation formula requires specific conditions on the function f to ensure the convergence of the infinite sums involved. For functions f in the [Schwartz class](/page/Schwartz_class) on \mathbb{R}^n, defined as infinitely differentiable functions with all derivatives satisfying sup_{x \in \mathbb{R}^n} |x|^k |\partial^\alpha f(x)| < \infty for all multi-indices \alpha and integers k \geq 0, both the left-hand side \sum_{m \in \mathbb{Z}^n} f(x + m) and the right-hand side \sum_{m \in \mathbb{Z}^n} \hat{f}(m) e^{2\pi i m \cdot x} converge absolutely and uniformly in x. This rapid decay property guarantees that the sums are well-defined pointwise without additional regularization.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf)[](https://www-users.cse.umn.edu/~garrett/m/mfms/notes_2013-14/02a_Poisson_summation.pdf) A weaker but sufficient condition for convergence is that f satisfies |f(x)| \leq C (1 + |x|)^{-N} for some constant C > 0 and N > n/2, where n is the dimension. This decay ensures the periodized sum converges in the L^1 sense, allowing the Fourier series expansion to represent the [periodization](/page/Periodization) accurately, though absolute convergence of the lattice sums may demand stronger decay such as N > n. The corresponding condition on \hat{f} is analogous, as the roles of f and \hat{f} are symmetric in the formula.[](https://www-users.cse.umn.edu/~garrett/m/mfms/notes_2013-14/02a_Poisson_summation.pdf) If the Fourier transform \hat{f} has compact support, the right-hand side sum reduces to a finite sum over the lattice points within that support, ensuring immediate convergence regardless of the decay of f itself. This scenario arises in bandlimited functions, where the formula equates the discrete samples to a finite Fourier series.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf) When absolute convergence fails, the formula can still hold under conditional convergence using methods like Cesàro or Abel summation. Cesàro summation averages partial sums to achieve convergence where ordinary sums oscillate, while Abel summation introduces a parameter, such as multiplying by r^{|m|} with r \to 1^-, to regularize the series. These techniques are essential for applications involving theta functions, where a Gaussian factor e^{-\pi t |m|^2} is added and the limit t \to 0^+ is taken to recover the original sum.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf) ### Extensions to Non-Schwartz Functions The Poisson summation formula, originally established for Schwartz functions, can be extended to certain classes of non-Schwartz functions under suitable convergence conditions. For functions $f \in L^1(\mathbb{R}^n)$ satisfying $|f(x)| \leq C(1 + |x|)^{-n - \delta}$ for some $\delta > 0$ and $C > 0$, along with the summability condition $\sum_{m \in \mathbb{Z}^n} |\hat{f}(m)| < \infty$, the formula holds in the form $\sum_{m \in \mathbb{Z}^n} \hat{f}(m) e^{2\pi i m \cdot x} = \sum_{k \in \mathbb{Z}^n} f(x + k)$ for all $x \in \mathbb{R}^n$.[](https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Grafakos_Classical_Fourier_Analysis.pdf) This extension relies on integration by parts or approximate identities to ensure the necessary decay and integrability, allowing the summation to converge absolutely on both sides.[](https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Grafakos_Classical_Fourier_Analysis.pdf) For compactly supported $C^\infty$ functions, the formula applies directly without additional restrictions beyond the support constraint. The Fourier transform $\hat{f}$ of such a function is an entire function of exponential type, ensuring rapid decay at infinity and enabling the Poisson summation to hold pointwise.[](https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Grafakos_Classical_Fourier_Analysis.pdf) This class includes test functions in $C_c^\infty(\mathbb{R}^n)$, where the compact support guarantees that only finitely many terms contribute in periodic summations, preserving the equality $\sum_{k \in \mathbb{Z}^n} f(k) = \sum_{m \in \mathbb{Z}^n} \hat{f}(m)$.[](https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Grafakos_Classical_Fourier_Analysis.pdf) To handle functions outside these classes, regularization techniques such as Gaussian smoothing are employed. Consider the smoothed version $f_t(x) = f(x) e^{-\pi t |x|^2}$ for $t > 0$, where the Poisson summation applies to the Schwartz function $f_t$, yielding $\sum_{m \in \mathbb{Z}} f_t(m) = \sum_{k \in \mathbb{Z}} \hat{f_t}(k)$. Taking the limit as $t \to 0^+$ recovers the original sums if they converge, providing a pathway for functions like those in $L^1$ with slower decay.[](https://www.math.columbia.edu/~woit/fourier-analysis/theta-zeta.pdf) In the multidimensional setting, the formula extends to general lattices $\Lambda \subset \mathbb{R}^n$ with dual lattice $\Lambda^* = \{ y \in \mathbb{R}^n \mid \langle x, y \rangle \in \mathbb{Z} \ \forall x \in \Lambda \}$. For a [Schwartz](/page/Schwartz) function $f$, it states $\sum_{x \in \Lambda} f(x) = \frac{1}{\mathrm{covol}(\Lambda)} \sum_{y \in \Lambda^*} \hat{f}(y)$, where $\mathrm{covol}(\Lambda)$ is the volume of the fundamental [parallelepiped](/page/Parallelepiped), adjusting for the lattice density.[](https://people.math.harvard.edu/~elkies/M272.19/sep16.pdf) This requires analogous conditions for non-Schwartz extensions, such as decay ensuring convergence of both lattice sums.[](https://people.math.harvard.edu/~elkies/M272.19/sep16.pdf) However, the formula fails for functions like polynomials or pure exponentials without principal part subtraction, as the sums diverge due to lack of decay. For instance, applying it naively to $f(x) = x^k$ or $f(x) = e^{i \alpha x}$ leads to infinite or oscillatory sums that do not match on both sides.[](https://www.mathoverflow.net/questions/14568/truth-of-the-poisson-summation-formula) ## Applications ### Method of Images in Electrostatics In electrostatics, the method of images is extended to periodic boundary conditions to model the potential generated by an infinite array of charges arranged on a lattice, such as in ionic crystals or periodic molecular systems. The charge density $\rho(\mathbf{r})$ is given by a periodic sum over the lattice points: $\rho(\mathbf{r}) = \sum_{\mathbf{m} \in \mathbb{Z}^n} q \delta(\mathbf{r} - \mathbf{m})$, where $q$ is the charge strength and $\mathbb{Z}^n$ denotes the $n$-dimensional integer lattice. The electrostatic potential $\phi(\mathbf{r})$ satisfies Poisson's equation $\nabla^2 \phi = -4\pi \rho$ (in Gaussian units), leading to a direct-space expression as the conditionally convergent lattice sum of Coulomb potentials: \phi(\mathbf{r}) = \sum_{\mathbf{m} \in \mathbb{Z}^n} \frac{q}{|\mathbf{r} - \mathbf{m}|}, which diverges without regularization due to the slow decay of the $1/r$ term.[](https://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf) To regularize this sum, the Poisson summation formula is applied by transforming to [Fourier](/page/Fourier) space. The [Fourier transform](/page/Fourier_transform) of the charge density yields $\hat{\rho}(\mathbf{k}) = q \sum_{\mathbf{l} \in \mathbb{Z}^n} e^{-2\pi i \mathbf{k} \cdot \mathbf{l}}$ for wavevectors $\mathbf{k}$ in the [reciprocal lattice](/page/Reciprocal_lattice). Solving [Poisson's equation](/page/Poisson's_equation) in reciprocal space gives $\hat{\phi}(\mathbf{k}) = \hat{\rho}(\mathbf{k}) / (4\pi |\mathbf{k}|^2)$ for $\mathbf{k} \neq 0$, with a separate constant term arising from the $\mathbf{k} = 0$ mode, often determined by neutrality conditions or [analytic continuation](/page/Analytic_continuation) to ensure convergence. The inverse [Fourier transform](/page/Fourier_transform) then expresses the potential as \phi(\mathbf{r}) = \frac{q}{V} + \sum_{\mathbf{k} \neq 0} \frac{q \sum_{\mathbf{l}} e^{2\pi i \mathbf{k} \cdot (\mathbf{r} - \mathbf{l})}}{4\pi |\mathbf{k}|^2} e^{2\pi i \mathbf{k} \cdot \mathbf{r}}, where $V$ is the unit cell volume; this reciprocal-space form converges rapidly due to the $1/|\mathbf{k}|^2$ decay.[](https://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf)[](https://arxiv.org/pdf/1409.3191) For practical evaluation, an Ewald-like splitting separates the potential into a short-range real-space sum (using a screened [Gaussian convergence factor](/page/Convergence)) and a long-range reciprocal-space sum, balancing computational efficiency and accuracy. The real-space part handles nearby images with exponential damping, while the [Fourier](/page/Fourier) part captures distant periodic contributions, avoiding the divergence of the unsplit sum. This approach effectively incorporates infinite images of the charge array to satisfy [periodic boundary conditions](/page/Periodic_boundary_conditions) without explicit truncation.[](https://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf)[](https://arxiv.org/pdf/1409.3191) Early applications of such lattice summation techniques in [electrostatics](/page/Electrostatics) date to the early [20th century](/page/20th_century), particularly in evaluating [crystal](/page/Crystal) potentials and Madelung constants for ionic [lattices](/page/Lattice), building on foundational work in [solid-state physics](/page/Solid-state_physics).[](https://www.davidhbailey.com/dhbpapers/PoissonLattice.pdf) ### Sampling and Nyquist-Shannon Theorem The Poisson summation formula establishes a direct link to the Nyquist-Shannon sampling theorem by enabling the exact reconstruction of bandlimited signals from their discrete samples, provided the sampling rate satisfies the [Nyquist criterion](/page/Nyquist_criterion). For a [function](/page/Function) $f$ whose [Fourier transform](/page/Fourier_transform) $\hat{f}$ is supported within $[-1/2, 1/2]$, the Poisson summation formula $\sum_{n \in \mathbb{Z}} f(n) = \sum_{k \in \mathbb{Z}} \hat{f}(k)$ simplifies significantly, as the support condition ensures $\hat{f}(k) = 0$ for all integers $k \neq 0$, yielding $\sum_{n \in \mathbb{Z}} f(n) = \hat{f}(0)$. This normalization corresponds to a Nyquist sampling rate of 1 (one sample per [unit interval](/page/Unit_interval)), where the [dual lattice](/page/Dual_lattice) aligns perfectly with the frequency support to avoid overlap.[](https://www-users.cse.umn.edu/~lerman/math5467/shannon_aliasing.pdf) Applying the Poisson formula to the periodization of $\hat{f}$, one obtains $\sum_{k \in \mathbb{Z}} \hat{f}(\xi + k) = \sum_{n \in \mathbb{Z}} f(n) e^{-2\pi i n \xi}$. Given the support in $[-1/2, 1/2]$, the left side equals $\hat{f}(\xi)$ within the fundamental period $[-1/2, 1/2]$, with no aliasing contributions from shifted copies.[](https://www-users.cse.umn.edu/~lerman/math5467/shannon_aliasing.pdf) Inverting the Fourier transform then yields the cardinal series interpolation formula for reconstruction: f(x) = \sum_{m \in \mathbb{Z}} f(m) \cdot \operatorname{sinc}(x - m), where $\operatorname{sinc}(y) = \sin(\pi y)/(\pi y)$. This series converges to $f(x)$ at every point, reproducing the original signal exactly from integer samples, and at sampling points $x = m$, it trivially satisfies $f(m) = \sum_{n} f(n) \operatorname{sinc}(m - n)$ due to the orthogonality of the sinc basis ($\operatorname{sinc}(m - n) = \delta_{mn}$). For general sampling rates, the formula extends via scaling: if samples are taken at intervals $T > 0$, the relevant Poisson summation involves the [lattice](/page/Lattice) $\mathbb{Z} T$ and dual frequency [lattice](/page/Lattice) $\mathbb{Z}/T$, with the Nyquist rate requiring $T \leq 1$ (or bandwidth $B \leq 1/(2T)$) to ensure the support of $\hat{f}$ fits within $[-1/(2T), 1/(2T)]$, preventing overlap in the periodized transform $\sum_{k} \hat{f}(\xi + k/T)$.[](https://www-users.cse.umn.edu/~lerman/math5467/shannon_aliasing.pdf) The reconstruction becomes $f(x) = \sum_{m} f(m T) \cdot \frac{\operatorname{sinc}((x - m T)/T)}{T}$, adjusted for the rate. Undersampling, where $T > 1/(2B)$, leads to [aliasing](/page/Aliasing) as the shifted copies in the dual sum $\sum_{k} \hat{f}(\xi + k/T)$ overlap within the [baseband](/page/Baseband), distorting the recoverable [spectrum](/page/Spectrum) and causing irreducible [information](/page/Information) loss in the [frequency domain](/page/Frequency_domain), a fundamental limit highlighted in [information theory](/page/Information_theory) contexts. This [aliasing](/page/Aliasing) manifests as spectral folding, where high-frequency components masquerade as lower ones, underscoring the theorem's role in determining minimal sampling requirements for faithful signal recovery. ### Ewald Summation in Molecular Dynamics In [molecular dynamics](/page/Molecular_dynamics) simulations of periodic systems, such as those modeling ionic crystals or biomolecular assemblies, the computation of long-range electrostatic interactions is hindered by the slow, [conditional convergence](/page/Conditional_convergence) of the infinite [lattice](/page/Lattice) sum $\sum_{\mathbf{n} \neq 0} 1/|\mathbf{r} + \mathbf{n}|$, where $\mathbf{n}$ runs over lattice vectors, leading to high computational cost and sensitivity to summation order.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1980.0135) The Ewald method, introduced by Paul Ewald in 1921, resolves this by screening each point charge $q_j$ at position $\mathbf{r}_j$ with a Gaussian function $\exp(-\alpha r^2)/r$, where $\alpha > 0$ controls the width, and splitting the potential into two parts: a short-range real-space sum involving the complementary error function $\mathrm{erfc}(\sqrt{\alpha} r)/r$ that decays rapidly and can be truncated at a finite cutoff, and a long-range reciprocal-space sum derived from the error function $\mathrm{erf}(\sqrt{\alpha} r)/r$.[](https://onlinelibrary.wiley.com/doi/10.1002/andp.19213690304) This decomposition ensures both sums converge quickly, with the optimal $\alpha$ balancing their computational efforts, typically yielding $O(N)$ scaling per timestep for $N$ particles after truncation.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1980.0135) The reciprocal-space contribution leverages the Poisson summation formula to express the periodic Gaussian potential as a Fourier series, resulting in the sum $\sum_{\mathbf{k} \neq 0} \frac{4\pi q_j}{k^2} \exp\left(-\frac{k^2}{4\alpha}\right) \exp(i \mathbf{k} \cdot \mathbf{r}_j)$, where $\mathbf{k}$ are reciprocal lattice vectors, providing an efficient evaluation via fast Fourier transforms in modern implementations.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1980.0135) To maintain charge neutrality and avoid artifacts, a self-interaction correction subtracts the Gaussian self-energy term $-\frac{2\alpha q_j^2}{\sqrt{\pi}}$ for each charge, while dipole moment corrections, such as those for tin-foil (metallic) or vacuum boundary conditions, adjust for the system's overall dipole to ensure physical accuracy in non-neutral or polarizable systems.[](https://pubs.aip.org/aip/jcp/article/155/11/114110/353403/Ewald-sum-corrections-in-simulations-of-ion-and) For large-scale simulations with thousands of particles, the standard Ewald method's $O(N^{3/2})$ reciprocal-space cost becomes prohibitive; the particle-mesh Ewald (PME) variant, developed in 1993, interpolates charges onto a uniform grid, solves the Poisson equation in reciprocal space using fast Fourier transforms, and back-interpolates forces, achieving $O(N \log N)$ scaling and widespread adoption in codes like [GROMACS](/page/GROMACS) and [AMBER](/page/AMBER) for biomolecular dynamics. ### Gaussian Integral Approximations The Poisson summation formula enables the approximation of integrals over $\mathbb{R}^n$ by sums over the [integer lattice](/page/Integer_lattice) $\mathbb{Z}^n$, particularly for rapidly decaying test functions where the error is quantified by the tails of the [dual lattice](/page/Dual_lattice) sum. For a [Schwartz](/page/Schwartz) function $f: \mathbb{R}^n \to \mathbb{C}$, the formula asserts that \sum_{m \in \mathbb{Z}^n} f(m) = \sum_{k \in \mathbb{Z}^n} \hat{f}(k), where $\hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i \langle x, \xi \rangle} \, dx$ is the [Fourier transform](/page/Fourier_transform) in the [standard](/page/Standard) convention. Since $\int_{\mathbb{R}^n} f(x) \, dx = \hat{f}(0)$, rearranging yields \sum_{m \in \mathbb{Z}^n} f(m) = \int_{\mathbb{R}^n} f(x) , dx + \sum_{k \in \mathbb{Z}^n \setminus {0}} \hat{f}(k). Thus, the [lattice](/page/Lattice) sum approximates the integral with error bounded by $\left| \sum_{k \neq 0} \hat{f}(k) \right|$, which decays rapidly if $\hat{f}$ is concentrated near the origin, as occurs for smooth, slowly varying $f$. This provides a rigorous framework for numerical approximations, with the [dual sum](/page/Sum) tails offering explicit error control.[](https://people.maths.ox.ac.uk/trefethen/sirev56-3_385.pdf) The [Gaussian function](/page/Gaussian_function) exemplifies this approximation due to its self-Fourier duality, leading to an exact identity via [Poisson](/page/Poisson) summation. Consider $f_\sigma(x) = e^{-\pi |x|^2 / \sigma}$ for $\sigma > 0$, whose [Fourier transform](/page/Fourier_transform) is $\hat{f}_\sigma(\xi) = \sigma^{n/2} e^{-\pi \sigma |\xi|^2}$. The [Poisson](/page/Poisson) formula then gives \sum_{m \in \mathbb{Z}^n} e^{-\pi |m|^2 / \sigma} = \sigma^{n/2} \sum_{k \in \mathbb{Z}^n} e^{-\pi \sigma |k|^2}. This is the multidimensional generalization of the Jacobi [theta function](/page/Theta_function) transformation law, where the left side is the [theta](/page/Theta) series $\theta(\mathbb{Z}^n; i/\sigma)$ and the right side reflects the modular inversion. For large $\sigma$, the original [sum](/page/Sum) approximates the [integral](/page/Integral) $\int f_\sigma = \sigma^{n/2}$ with exponentially small [error](/page/Error) from the dual tails: $\left| \sum_{k \neq 0} \hat{f}_\sigma(k) \right| \leq 2^{n+1} \sigma^{n/2} e^{-\pi \sigma}$ (bounding over the nearest [lattice](/page/Lattice) points and using [Gaussian](/page/Gaussian_function) decay), yielding an $O(\sigma^{n/2} e^{-\pi \sigma})$ [error](/page/Error) term as $\sigma \to \infty$. Such identities facilitate volume approximations in [lattice](/page/Lattice)-based computations, like estimating the [content](/page/Content) of [fundamental](/page/Fundamental) domains. Higher accuracy can be achieved by refining the approximation with the Euler-Maclaurin formula, which introduces corrections via [Bernoulli polynomials](/page/Bernoulli_polynomials) to capture boundary effects in the lattice sum. The Euler-Maclaurin expansion relates sums to integrals plus terms involving derivatives of $f$ weighted by Bernoulli numbers, and when combined with Poisson summation, it yields an asymptotic series for the error, such as $\sum_m f(m) - \int f = \sum_{k \neq 0} \hat{f}(k) + \sum_{j=1}^p \frac{B_{2j}}{(2j)!} (\hat{f}^{(2j-1)}(0) - f^{(2j-1)}(0)) + O(\|f^{(2p+1)}\|)$, where $B_{2j}$ are Bernoulli numbers. This hybrid approach enhances precision for non-Gaussian functions by incorporating polynomial corrections alongside the exponential dual tails, often simplifying computations in scenarios where direct Euler-Maclaurin application is cumbersome. In numerical [quadrature](/page/Quadrature) over $\mathbb{R}^n$, [Poisson](/page/Poisson) summation underpins lattice-based methods like the multidimensional [trapezoidal rule](/page/Trapezoidal_rule), offering exponential convergence for rapidly decaying integrands as a deterministic alternative to [Monte Carlo](/page/Monte_Carlo) sampling, which typically yields slower $O(1/\sqrt{N})$ rates. For instance, in one dimension, approximating $\int_{-\infty}^\infty g(x) \, dx$ by $h \sum_k g(k h)$ incurs discretization error $O(e^{-c / h^2})$ via [Poisson](/page/Poisson) analysis of the [Fourier](/page/Fourier) coefficients, with $c = \pi^2$ for Gaussian $g(x) = e^{-x^2}$; optimal $h \sim 1/\sqrt{n}$ (for $n$ points) achieves $O(e^{-\pi n})$ accuracy. Extensions to $\mathbb{R}^n$ employ tensor-product [lattices](/page/Lattice) or Voronoi cells, partitioning the sum into real-space and reciprocal-space contributions for efficient evaluation of Gaussian integrals in periodic settings, with errors decaying exponentially in the lattice density. These techniques are particularly effective for high-dimensional problems in physics and [engineering](/page/Engineering), where [Monte Carlo](/page/Monte_Carlo) variance is prohibitive.[](https://people.maths.ox.ac.uk/trefethen/sirev56-3_385.pdf) ### Counting Lattice Points in Regions The Gauss circle problem seeks to determine the number of points of the integer lattice $\mathbb{Z}^2$ lying inside or on the boundary of a disk of radius $R$ centered at the origin, denoted $N(R)$. This count satisfies $N(R) = \pi R^2 + E(R)$, where $E(R)$ represents the error term measuring the discrepancy between the discrete sum and the continuous volume.[](https://arxiv.org/pdf/math/0304187) Applying the Poisson summation formula to a smoothed version of the indicator function of the disk transforms the lattice sum into an integral plus a dual sum over the Fourier transform, yielding an explicit expression for $E(R)$ involving oscillatory terms. Specifically, for a smoothed approximation, the error takes the form $\sqrt{X} \sum_{n \geq 1} n^{-1/2} r_2(n) \psi(\delta \sqrt{n}) J_1(2\pi \sqrt{n X})$, where $X = R^2$, $r_2(n)$ is the number of ways to write $n$ as a sum of two squares, $\psi$ is a smoothing function, $\delta = X^{-1/6}$, and $J_1$ is the Bessel function of the first kind; the asymptotic behavior of the Bessel function produces oscillations on the order of $\sum r^{-1/2}$. This approach, originally developed by Voronoi, provides the bound $E(R) = O(R^{2/3 + \epsilon})$ for any $\epsilon > 0$.[](https://arxiv.org/pdf/math/0304187) In a more general setting, the Poisson summation formula applied to the [indicator function](/page/Indicator_function) $\chi_D$ of a bounded [domain](/page/Domain) $D \subset \mathbb{R}^d$ gives $\sum_{m \in \mathbb{Z}^d} \chi_D(m) \approx \mathrm{vol}(D) + \sum_{k \in \mathbb{Z}^d \setminus \{0\}} \hat{\chi}_D(k)$, where $\hat{\chi}_D$ denotes the [Fourier transform](/page/Fourier_transform); for non-smooth $\chi_D$, this is understood via [approximation](/page/Approximation) by test functions to handle [convergence](/page/Convergence). For spherical regions, the Hardy-Littlewood method exploits the radial symmetry by computing the [Fourier transform](/page/Fourier_transform) of radial functions, which involves [Bessel functions](/page/Bessel_function), to estimate the dual sum and derive the oscillatory contributions to the error.[](https://arxiv.org/pdf/math/0304187) The conjectured optimal error bound for the Gauss circle problem is $E(R) = O(R^{1/2 + \epsilon})$ for any $\epsilon > 0$, reflecting the $\Omega(R^{1/2})$ lower bound established by Hardy and Landau through the explicit oscillatory terms from Poisson summation; while this remains open, the formula provides the mechanism for these oscillations and informs subconvexity efforts. In higher dimensions, the lattice point discrepancy for balls of radius $R$ in $\mathbb{R}^d$ follows an analogous structure, with Poisson summation yielding error terms of size $O(R^{d-2 + \epsilon})$ conjectured optimally, and explicit dual sums linking to the geometry of the lattice. These discrepancies connect to Epstein zeta functions $\zeta_Q(s) = \sum_{m \in \mathbb{Z}^d \setminus \{0\}} Q(m)^{-s}$ for positive definite quadratic forms $Q$, as the theta series $\theta_Q(\tau) = \sum_{m \in \mathbb{Z}^d} e^{\pi i \tau Q(m)}$ associated with lattice point enumeration satisfies a functional equation via Poisson summation, and the Mellin transform relates it to $\zeta_Q(s)$; this framework facilitates asymptotic analysis and moment estimates for random lattices in large dimensions.[](https://www.sciencedirect.com/science/article/pii/S0001870819300702)[](https://www.reed.edu/physics/faculty/crandall/papers/epstein.pdf) ### Analytic Number Theory Results The Poisson summation formula plays a pivotal role in [analytic number theory](/page/Analytic_number_theory) by facilitating the transformation of sums over arithmetic functions into dual sums that reveal asymptotic behaviors and error terms. It is instrumental in deriving precise estimates for sums involving multiplicative functions, such as the [divisor function](/page/Divisor_function), and in establishing functional equations for L-functions. These applications often exploit the formula's ability to relate [lattice](/page/Lattice) sums to their [Fourier](/page/Fourier) transforms, enabling the analysis of oscillatory phenomena and [spectral](/page/Spectral) contributions in number-theoretic contexts.[](https://arxiv.org/pdf/math/0304187) In the Dirichlet divisor problem, the Poisson summation formula is applied to the Fourier transform of the divisor function $d(n)$, which counts the number of positive divisors of $n$. The partial sum $\sum_{n \leq x} d(n)$ admits the asymptotic expansion $x \log x + (2\gamma - 1)x + \Delta(x)$, where $\gamma$ is the Euler-Mascheroni constant and $\Delta(x)$ denotes the error term. By invoking Poisson summation on the dual sum involving the Fourier coefficients of $d(n)$, one obtains oscillatory contributions of size approximately $\sqrt{x}$ from the zeros of the Riemann zeta function, highlighting the formula's role in capturing the secondary main term and bounding the error. This approach, originating in classical treatments, underscores the interplay between additive and multiplicative structures in arithmetic sums.[](https://arxiv.org/pdf/math/0304187) The Voronoi summation formula extends the Poisson summation to automorphic forms on $\mathrm{GL}(2)$ over number fields, providing a powerful tool for summing Fourier coefficients twisted by additive characters. Specifically, for a cusp form $\phi$ on $\mathrm{GL}(2, \mathbb{Q})$, the formula transforms $\sum_{n \leq X} a_\phi(n) e( n \alpha ) W(n/X)$ into a dual sum involving Kloosterman sums $ \mathrm{Kl}(m, n; c) = \sum_{d \bmod c}^* e( (m \bar{d} + n d)/c ) $, weighted by the Whittaker function and Bessel terms. This generalization, derived via the representation theory of adelic groups, is essential for evaluating sums over Kloosterman sums and has applications in bounding moments of L-functions associated to these forms. The formula's structure mirrors the Poisson case but incorporates the archimedean and non-archimedean components, enabling precise control over spectral parameters.[](https://arxiv.org/abs/1410.3410) A foundational application appears in the proof of the [functional equation](/page/Functional_equation) for the Riemann zeta function $\zeta(s)$, achieved through Poisson summation applied to the [Gaussian function](/page/Gaussian_function). Define the [theta function](/page/Theta_function) $\theta(\tau) = \sum_{n \in \mathbb{Z}} e^{ \pi i n^2 \tau }$ for $\operatorname{Im} \tau > 0$. The Poisson summation formula yields the transformation law $\theta(\tau) = \tau^{-1/2} \theta(1/\tau)$, which, upon Mellin inversion, relates $\zeta(s)$ to $\zeta(1-s)$ via the completed function $\Lambda(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s)$, satisfying $\Lambda(s) = \Lambda(1-s)$. This derivation, central to [analytic continuation](/page/Analytic_continuation) and the [prime number theorem](/page/Prime_number_theorem), demonstrates how Poisson summation bridges the [Dirichlet series](/page/Dirichlet_series) representation with its [analytic continuation](/page/Analytic_continuation).[](https://www-users.cse.umn.edu/~garrett/m/mfms/notes_2013-14/02_Riemann_and_zeta.pdf) For Dirichlet L-functions $L(s, \chi)$ attached to primitive characters $\chi \bmod q$, the approximate functional equation expresses $L(1/2 + it, \chi)$ as a finite sum $\sum_{n \leq \sqrt{q(1+|t|)}} \chi(n) n^{-1/2 - it} V(n)$ plus a dual sum over the characters, truncated at similar length. Poisson summation is invoked on the generating function for the characters, transforming the sum into contributions from the dual lattice $\mathbb{Z}/q\mathbb{Z}^\vee$, which incorporates Gauss sums and reveals the role of the root number in the functional equation. This formulation, crucial for computing central values and moments, facilitates the study of distribution in arithmetic progressions via spectral methods.[](https://arxiv.org/pdf/2510.02273) In modern developments, Poisson summation underpins subconvexity bounds for L-functions through its integration with [spectral theory](/page/Spectral_theory) on automorphic forms. For instance, in the GL(3) × GL(2) Rankin-Selberg setting, applying Poisson to the oscillator sum in the [delta method](/page/Delta_method) yields subconvex estimates like $L(1/2, \phi \times f) \ll (q t)^{1/2 - \delta}$ for conductor $q t$, where $\phi$ is a GL(3) form and $f$ a GL(2) form, with $\delta > 0$ depending on the [spectral](/page/Spectral) aspect. These bounds, leveraging Voronoi-type dual sums to control hyper-Kloosterman contributions, advance toward the [Lindelöf hypothesis](/page/Lindelöf_hypothesis) and have implications for moments and zero statistics in the critical strip.[](https://arxiv.org/pdf/2006.07819) ### Sphere Packings and Crystallography In the context of sphere packings, the Poisson summation formula facilitates the analysis of packing densities through lattice theta functions. For an $n$-dimensional lattice $\Lambda \subset \mathbb{R}^n$, the theta function is given by \theta_\Lambda(\tau) = \sum_{m \in \Lambda} e^{\pi i \tau |m|^2}, where $\Im(\tau) > 0$. The Poisson summation formula establishes the functional equation \theta_\Lambda(\tau) = (\det \Lambda)^{-1/2} \theta_{\Lambda^*}(-1/\tau), relating $\theta_\Lambda$ to the theta function of the dual lattice $\Lambda^*$. This duality encodes the distribution of lattice vectors by squared lengths, with the constant term $\theta_\Lambda(i\infty) = 1$ and subsequent coefficients determining the number of minimal vectors, which directly influences the packing radius and density $\delta(\Lambda) = (\det \Lambda)^{-1/2} / 2^{\mu/2}$, where $\mu$ is the minimal norm. The relation aids in comparing packings by transforming sums over short vectors to those over the dual, optimizing global density.[](https://people.math.harvard.edu/~elkies/M272.19/sep16.pdf) A representative example is the face-centered cubic (FCC) lattice in three dimensions, which realizes the maximal packing density $\delta = \pi / (3\sqrt{2}) \approx 0.7405$. Its theta function $\theta_{\mathrm{FCC}}(\tau)$ satisfies the Poisson duality with the dual body-centered cubic lattice, equating the series expansion over FCC points to a scaled dual sum, thereby confirming density maximization through the absence of shorter non-zero vectors in the transformed series. This equivalence highlights how Poisson summation verifies optimality without exhaustive enumeration.[](https://d-nb.info/1208292358/34) Siegel theta series extend these concepts to higher-genus modular forms associated with [quadratic form](/page/Quadratic_form)s, leveraging Poisson summation for their transformation properties. For a positive definite even [integral](/page/Integral) [quadratic form](/page/Quadratic_form) $Q$ of dimension $m$, the Siegel theta series of [genus](/page/Genus) $g$ is \theta_Q^{(g)}(Z) = \sum_{T \in \mathrm{M}_g(\mathbb{Z})} r_Q(T) e^{\pi i \mathrm{tr}(T Z)}, where $Z \in \mathbb{H}_g$ (Siegel upper half-space) and $r_Q(T)$ counts representations of matrix $T$ by $Q$. Poisson summation derives the automorphy factor: under $\gamma = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \mathrm{Sp}_{2g}(\mathbb{Z})$, \theta_Q^{(g)}(\gamma Z) = \det(CZ + D)^{m/2} \theta_Q^{(g)}(Z), making $\theta_Q^{(g)}$ a Siegel modular form of weight $m/2$. This framework analyzes multi-dimensional [lattice](/page/Lattice) packings by linking representation numbers to modular invariance, essential for classifying dense configurations.[](https://www.mathi.uni-heidelberg.de/~winfried/siegel2.pdf) In [crystallography](/page/Crystallography), Poisson summation governs summations over [space group](/page/Space_group)s, interpreting [Fourier](/page/Fourier) coefficients as structure factors that model atomic scattering amplitudes. For a crystal [lattice](/page/Lattice) $\Gamma$, the [electron density](/page/Electron_density) $\rho(x) = \sum_{g \in G} f_g \delta(x - g)$, where $G$ is the [space group](/page/Space_group) and $f_g$ atomic form factors, yields structure factors $F(h) = \sum_{g \in G} f_g e^{2\pi i \langle h, g \rangle}$ via the [Fourier transform](/page/Fourier_transform). The [formula](/page/Formula) equates the direct [lattice](/page/Lattice) [sum](/page/Sum) to a [reciprocal lattice](/page/Reciprocal_lattice) [sum](/page/Sum), producing Bragg peaks in [diffraction](/page/Diffraction) patterns and enabling [phase retrieval](/page/Phase_retrieval) for structure determination from intensities. This duality is foundational for [X-ray crystallography](/page/X-ray_crystallography), distinguishing periodic crystals by pure-point spectra.[](https://www.sciencedirect.com/science/article/pii/S0393044006001471) The Kabatiansky-Levenshtein bound utilizes Poisson summation to establish rigorous upper limits on [sphere packing](/page/Sphere_packing) densities in high dimensions, focusing on spherical codes and [lattice](/page/Lattice) theta series. By applying the formula to radial functions $f$ with $f \geq 0$ near the [origin](/page/Origin) and non-positive elsewhere, and ensuring $\hat{f} \geq 0$, the [density](/page/Density) $\Delta_n$ satisfies $\Delta_n \leq 2^{-0.599n + [o(n](/page/O(n))}$, derived from optimizing the dual sum over the [reciprocal lattice](/page/Reciprocal_lattice) to constrain vector distributions. This bound, improving Minkowski-Hlawka asymptotics, relies on the theta series [integral](/page/Integral) representation $\int \theta_\Lambda(z) f(\|x\|) dx = (\det \Lambda)^{-1/2} \int \theta_{\Lambda^*}(z) \hat{f}(\|y\|) dy$, highlighting unattainability of denser packings beyond certain dimensions.[](https://annals.math.princeton.edu/wp-content/uploads/annals-v157-n2-p09.pdf) ### Uncertainty Principle Connections The Donoho–Stark uncertainty principle provides a framework for understanding the trade-offs in the concentration of a function $f$ and its Fourier transform $\hat{f}$ on measurable sets, with direct implications for the Poisson summation formula. Specifically, for $f \in L^2(\mathbb{R})$ that is $\epsilon$-concentrated on a set $T \subset \mathbb{R}$ of finite measure $|T|$ (meaning $\int_{|x| \notin T} |f(x)|^2 \, dx \leq \epsilon \|f\|_2^2$) and $\hat{f}$ $\eta$-concentrated on a set $F \subset \mathbb{R}$ of finite measure $|F|$, the principle states that $|T| \cdot |F| \geq (1 - \epsilon - \eta)^2$, assuming the Fourier transform is normalized so that the constant is 1. In the discrete setting relevant to lattice sums, for $f: \mathbb{Z} \to \mathbb{C}$ with $\sum_{m \in \mathbb{Z}} |f(m)|^2 > 0$ and $\sum_{k \in \mathbb{Z}} |\hat{f}(k)|^2 > 0$, where $\hat{f}(k) = \sum_{m} f(m) e^{-2\pi i k m}$, the supports satisfy $|\operatorname{supp} f| \cdot |\operatorname{supp} \hat{f}| \geq |\mathbb{Z}|$, but since $\mathbb{Z}$ is infinite, this implies that both supports cannot be finite unless $f \equiv 0$. This bound arises from the incompatibility of finite spatial and frequency supports, mirroring the classical Heisenberg uncertainty but quantified for $L^2$-norm concentrations. The Poisson summation formula, $\sum_{m \in \mathbb{Z}} f(m) = \sum_{k \in \mathbb{Z}} \hat{f}(k)$, links these sums directly to the uncertainty trade-off: if $f$ is well-localized in the spatial [domain](/page/Domain) (small effective [support](/page/Support) for the left sum to converge rapidly), then $\hat{f}$ must spread out in the [frequency domain](/page/Frequency_domain) (delocalized right sum), and vice versa, to ensure both sides converge. Gröchenig's [uncertainty principle](/page/Uncertainty_principle) extends this by establishing conditions under which an $L^p$-$L^q$ [inequality](/page/Inequality), such as $\left( \sum_m |f(a m + b)|^p \right)^{1/p} \left( \sum_k |\hat{f}(c k + d)|^q \right)^{1/q} \geq C > 0$ for parameters $a, b, c, d, p, q$, guarantees [absolute convergence](/page/Absolute_convergence) of both sums in the Poisson formula. Violation of such inequalities implies divergence, reinforcing that tight localization in one domain prevents the formula's validity with absolute convergence. This connection highlights how Poisson summation enforces delocalization for practical computability in sampled signals, akin to bandlimited sampling limits but deriving fundamental bounds rather than reconstruction methods. The Balian–Low theorem further exemplifies this duality, proving that no nonzero $g \in L^2(\mathbb{R})$ can satisfy both $\int_{\mathbb{R}} |x|^2 |g(x)|^2 \, dx < \infty$ (finite time variance) and $\int_{\mathbb{R}} |\xi|^2 |\hat{g}(\xi)|^2 \, d\xi < \infty$ (finite frequency variance) while generating a Riesz basis via the Gabor system $\{ e^{2\pi i k x} g(x - m) \}_{k,m \in \mathbb{Z}}$ for $L^2(\mathbb{R})$. The proof relies on Poisson summation applied to the Zak transform of $g$, which reveals that finite variances would imply a contradiction in the basis expansion, as the dual lattice contributions force at least one variance to be infinite. Quantitatively, this yields lower bounds on the variances from [dual lattice](/page/Dual_lattice) terms; for instance, if the time variance is finite, the frequency variance satisfies $\int |\xi|^2 |\hat{g}(\xi)|^2 \, d\xi \geq C \left( \sum_{k \in \mathbb{Z} \setminus \{0\}} \frac{1}{k^2} \right)^{-1}$ for some constant $C > 0$, derived from the Poisson-dual spreading and noting that $\sum_{k \neq 0} \frac{1}{k^2} = \frac{\pi^2}{3}$. In [quantum mechanics](/page/Quantum_mechanics), this analogy manifests in the position-momentum uncertainty for wave packets on lattices, where Weyl quantization sums over phase-space points enforce $\Delta x \Delta p \geq \hbar/2$, with Poisson summation quantifying the minimal spread via [dual lattice](/page/Dual_lattice) interference. ## Generalizations ### Selberg Trace Formula The [Selberg trace formula](/page/Selberg_trace_formula) represents a non-abelian [generalization](/page/Generalization) of the [Poisson summation formula](/page/Poisson_summation_formula), extending its principles from [Euclidean](/page/Euclidean) lattices to the [spectral](/page/Spectral) [geometry](/page/Geometry) of [hyperbolic](/page/Hyperbolic) surfaces. In this context, it equates a spectral sum over eigenvalues of the Laplace-Beltrami operator to a geometric sum over lengths of closed geodesics, providing a duality between the spectrum and the topology of the manifold. This formula has profound implications in [analytic number theory](/page/Analytic_number_theory), particularly for studying automorphic forms and [zeta](/page/Zeta) functions on spaces like the modular surface. The setup is typically formulated on the quotient space $\Gamma \backslash G$, where $G = \mathrm{SL}(2, \mathbb{R})$ acts on the hyperbolic plane $\mathbb{H}^2$, and $\Gamma = \mathrm{SL}(2, \mathbb{Z})$ is a [discrete](/page/Discrete) subgroup, yielding the non-compact modular surface $\mathrm{SL}(2, \mathbb{Z}) \backslash \mathbb{H}^2$. The hyperbolic Laplacian $\Delta$ on this space has a [discrete](/page/Discrete) spectrum of eigenvalues $\lambda_n = s_n(1 - s_n)$, where $s_n = \frac{1}{2} + i r_n$ for $r_n \geq 0$, corresponding to Maass cusp forms or [Eisenstein series](/page/Eisenstein_series). The trace formula arises from the [spectral decomposition](/page/Spectral_decomposition) of invariant kernels under the [group action](/page/Group_action), mirroring how the Poisson formula decomposes sums over a [lattice](/page/Lattice) into dual [Fourier](/page/Fourier) sums.[](https://people.maths.bris.ac.uk/~majm/bib/selberg.pdf) In its distributional form, the Selberg trace formula relates spectral distributions to orbital integrals, but explicit delta function expressions vary by presentation. For a more regularized version using a test function $h(r)$ even and decreasing, the formula becomes \sum_j h(r_j) = \frac{\mathrm{Area}(M)}{4\pi} \int_{-\infty}^\infty h(r) \frac{\tanh(\pi r)}{r} , dr + \sum_{\gamma \in P} \sum_{k=1}^\infty \frac{\ell(\gamma) \hat{h}(k \ell(\gamma))}{2 \sinh(k \ell(\gamma)/2)}, where $P$ is the set of primitive hyperbolic conjugacy classes, $r_j = \sqrt{\lambda_j - 1/4}$, and $\hat{h}$ is the [Fourier transform](/page/Fourier_transform) of $h$. The origin in Poisson summation lies in the spectral decomposition of the delta function on the group, which is dual to sums over orbital integrals along conjugacy classes, generalizing the lattice point counting in the abelian case of flat tori.[](https://people.maths.bris.ac.uk/~majm/bib/selberg.pdf)[](https://static.renyi.hu/~major/articles/selberg1.pdf) An explicit realization uses the [heat kernel](/page/Heat_kernel) $K_t(x, y)$, the fundamental solution to the [heat equation](/page/Heat_equation) $(\partial_t + \Delta) K_t = 0$ on the manifold, which admits a [spectral](/page/Spectral) expansion $K_t(x, y) = \sum_n e^{-\lambda_n t} \phi_n(x) \phi_n(y)$ and a geometric expansion via images under $\Gamma$. The [trace](/page/Trace) [formula](/page/Formula) then gives \int_M K_t(x, x) , d\mu(x) = \sum_n e^{-\lambda_n t} + \sum_{\gamma \in \Gamma \setminus {e}} \int_{M/\mathrm{Cent}(\gamma)} K_t(x, \gamma x) , d\mu_\gamma(x), where the left side is the [spectral](/page/Spectral) [trace](/page/Trace), the right includes the identity contribution plus sums over non-trivial conjugacy classes, with integrals over centralizers $\mathrm{Cent}(\gamma)$. This [heat kernel](/page/Heat_kernel) form directly parallels the [theta function](/page/Theta_function) identity in the classical [Poisson](/page/Poisson) [formula](/page/Formula) for the [torus](/page/Torus).[](https://people.maths.bris.ac.uk/~majm/bib/selberg.pdf)[](https://static.renyi.hu/~major/articles/selberg1.pdf) Applications of the Selberg trace formula include the prime geodesic theorem, which asserts that the number $\pi(x)$ of primitive closed geodesics of length at most $x$ on [the surface](/page/The_Surface) satisfies $\pi(x) \sim \frac{x}{2\pi} \log \frac{x}{2\pi}$ as $x \to \infty$, analogous to the [prime number theorem](/page/Prime_number_theorem). This is derived by applying the trace formula to suitable test functions and analyzing the error terms via the Selberg zeta function, whose zeros interlace the spectral eigenvalues and geodesic lengths. The formula also enables counting "primes" (primitive geodesics) through the distribution of these zeros, providing explicit error bounds and connections to the [Riemann hypothesis](/page/Riemann_hypothesis) for the zeta function.[](https://people.maths.bris.ac.uk/~majm/bib/selberg.pdf) Historically, [Atle Selberg](/page/Atle_Selberg) introduced the trace formula in 1956, building on the classical Poisson summation formula for flat tori by replacing [abelian group](/page/Abelian_group) sums with traces of operators on non-compact Riemannian manifolds of constant negative curvature. This work, presented in his address to the International Colloquium on Automorphic Functions, unified [harmonic analysis](/page/Harmonic_analysis) and geometry, influencing subsequent developments in [spectral theory](/page/Spectral_theory) and the [Langlands program](/page/Langlands_program). ### Semiclassical Trace Formula The semiclassical trace formula extends the principles of the Poisson summation formula to [quantum systems](/page/Quantum-Systems) whose classical counterparts exhibit [chaotic](/page/Chaos) dynamics, establishing a duality between quantum energy levels and classical periodic orbits in the [limit](/page/Limit) as Planck's constant $\hbar$ approaches zero. This framework approximates the quantum spectral properties, such as the [density of states](/page/Density_of_states) or the [trace](/page/Trace) of the time-evolution operator, through sums over unstable periodic orbits, thereby bridging [quantum mechanics](/page/Quantum_mechanics) and classical [chaos](/page/Chaos). The analogy to [Poisson](/page/Poisson) summation lies in the [decomposition](/page/Decomposition) of the quantum [trace](/page/Trace) into a smooth background term (capturing average spectral behavior) and an oscillating correction (encoding dynamical details from periodic orbits), much like how [Poisson](/page/Poisson) summation separates a function's [direct sum](/page/Direct_sum) from its dual [Fourier](/page/Fourier) contributions.[](http://www.lptms.u-psud.fr/membres/ullmo/Articles/eolss-ullmo-tomsovic.pdf) The cornerstone of this approach is the Gutzwiller trace formula, which provides a semiclassical expression for the trace of the quantum propagator. For a time-independent Hamiltonian $H$, it states that \operatorname{Tr} \left( e^{-i H t / \hbar} \right) \approx \sum_{\gamma} A_{\gamma} , e^{i S_{\gamma} / \hbar - i \mu_{\gamma} \pi / 2}, where the sum runs over primitive periodic orbits $\gamma$, $S_{\gamma}$ denotes the classical action along orbit $\gamma$, $\mu_{\gamma}$ is the Maslov index counting caustics encountered, and $A_{\gamma}$ is a prefactor involving the orbit's monodromy matrix and period, reflecting its stability. This approximation holds in the semiclassical regime $\hbar \to 0$ for systems with hyperbolic dynamics, where short orbits dominate the sum. In this formula, the role of Poisson summation manifests through the classical sum over orbits acting as a [dual representation](/page/Dual_representation) to the quantum eigenvalues, with the smooth component governed by the [Weyl law](/page/Weyl_law)—expressing the average [density of states](/page/Density_of_states) as the phase-space volume below energy $E$ divided by $(2\pi \hbar)^f$ for $f$ [degrees of freedom](/page/Degrees_of_freedom). The Weyl term provides the leading semiclassical correction akin to the zero-frequency term in Poisson summation, while the orbit contributions introduce fluctuations that reveal the system's [chaotic](/page/Chaotic) structure. For integrable systems, where chaos is absent, the Berry-Tabor [conjecture](/page/Conjecture) modifies this picture: the semiclassical trace involves sums over quantized invariant tori (corresponding to action-angle variables) rather than isolated orbits, leading to uncorrelated eigenvalue spacings that follow a [Poisson distribution](/page/Poisson_distribution), in contrast to the level repulsion seen in chaotic cases. These trace formulas find applications in analyzing eigenvalue statistics for quantum [billiards](/page/Billiard), where chaotic billiard shapes yield spectral correlations matching random matrix theory predictions (e.g., Gaussian orthogonal [ensemble](/page/Ensemble!) for time-reversal symmetric systems), while integrable shapes align with the [Poisson](/page/Poisson) statistics of the Berry-Tabor [conjecture](/page/Conjecture). They also explain quantum scarring, a phenomenon in which certain eigenstates concentrate probability density along unstable classical periodic orbits in otherwise ergodic systems, as observed in numerical studies of [stadium](/page/Stadium) and [Sinai](/page/Sinai) [billiards](/page/Billiard). A key connection to spectral geometry arises from viewing the length spectrum—the Fourier transform of the density of states with respect to [energy](/page/Energy)—as directly encoding the lengths of [primitive](/page/Primitive) periodic orbits through prominent peaks, allowing [extraction](/page/Extraction) of classical dynamical [information](/page/Information) from quantum spectra in bounded domains. ### Poisson Summation on Lie Groups The Poisson summation formula admits a [generalization](/page/Generalization) to [Lie](/page/Lie) groups, extending the classical abelian case to non-abelian structures through the framework of unitary [representation theory](/page/Representation_theory). In the abelian setting, consider the quotient $\mathbb{R}/G$ where $G$ is a compact abelian [Lie](/page/Lie) group, such as a [torus](/page/Torus); here, the formula relates the sum of a suitable function $f$ over the lattice points induced by $G$ to the sum of its [Fourier](/page/Fourier) coefficients on the Pontryagin [dual](/page/Dual) group $\hat{G}$, which is [discrete](/page/Discrete). This setup captures periodic summation akin to the standard $\mathbb{R}^n/\mathbb{Z}^n$ case, with convergence ensured for Schwartz-class functions. The extension to compact non-abelian [Lie](/page/Lie) groups replaces the Pontryagin [dual](/page/Dual) with the [discrete](/page/Discrete) set of equivalence classes of irreducible unitary representations $[\pi] \in \hat{G}$, where the [Fourier transform](/page/Fourier_transform) of $f \in L^1(G)$ is given by the operator-valued coefficients $\hat{f}(\pi) = \int_G f(g) \pi(g)^* dg$, with $dg$ the normalized [Haar measure](/page/Haar_measure). For a general compact [Lie group](/page/Lie_group) $G$ and a closed [subgroup](/page/Subgroup) $H$, the formula applies to homogeneous spaces $G/H$ with $G$-invariant measure $\mu$. The Poisson [summation](/page/Summation) equates integrals over $G/H$ to sums over the representation-theoretic dual $\hat{G/H} = \{[\pi] \in \hat{G} : \pi|_H$ contains the trivial representation\}\). Specifically, for $f \in C(G)$, the main identity is \int_{G/H} f(xH) d\mu(xH) = \sum_{[\pi] \in \hat{G/H}} d_\pi \operatorname{tr} \left[ \hat{f}(\pi) (T_H^\pi)^* \right], where $d_\pi = \dim \pi$ is the dimension of $\pi$, $\hat{f}(\pi)$ is the [Fourier](/page/Fourier) coefficient, and $T_H^\pi$ projects onto the $H$-[invariant](/page/Invariant) subspace. This generalizes the abelian case by using traces of representation operators instead of characters, enabling [summation](/page/Summation) over coadjoint orbits or lattices in the dual. The inversion follows from the [Plancherel theorem](/page/Plancherel_theorem) for compact groups, which asserts \int_{G/H} | \varphi(xH) |^2 d\mu(xH) = \sum_{[\pi] \in \hat{G/H}} d_\pi | \hat{\varphi}(\pi) |_{HS}^2, with $\| \cdot \|_{HS}$ the Hilbert-Schmidt norm, providing a direct means to recover $\varphi$ from its [Fourier](/page/Fourier) data.[](https://link.springer.com/article/10.1007/s13324-016-0156-2) In the non-compact case, for semisimple [Lie group](/page/Lie_group)s, Harish-Chandra's Plancherel formula replaces the discrete sum with an integral over the continuous parameter space of unitary representations, supported on the tempered spectrum. For a real semisimple [Lie group](/page/Lie_group) $G$ with maximal compact [subgroup](/page/Subgroup) $K$, the Plancherel measure $\mu_G$ decomposes $L^2(G)$ as an integral over irreducible representations $\pi_\lambda$ parameterized by $\lambda \in \mathfrak{a}^*$ (the dual of a [Cartan subalgebra](/page/Cartan_subalgebra)), weighted by the formal degree and Harish-Chandra's $c$-function: \int_G |f(g)|^2 dg = \int_{\hat{G}} | \hat{f}(\pi) |_{HS}^2 d\mu_G(\pi), where the inversion yields the Poisson-type relation for sums over discrete [subgroup](/page/Subgroup)s $\Gamma \subset G$ via the spectral expansion on $\Gamma \backslash G$. This structure inverts the [Fourier transform](/page/Fourier_transform), linking [lattice](/page/Lattice) sums to integrals over the unitary dual. For instance, on non-compact homogeneous spaces like [hyperbolic space](/page/Hyperbolic_space) $\mathbb{H}^n = SO(n,1)/SO(n)$, the formula manifests at the group level through the continuous series representations, providing a [spectral decomposition](/page/Spectral_decomposition) that connects to automorphic forms without invoking trace identities. Examples illustrate these generalizations. On the compact group $SU(2)$, the irreducible representations are labeled by half-integers $j = 0, 1/2, 1, \dots$, with dimensions $d_j = 2j+1$; the Poisson summation for functions on $SU(2)/U(1) \cong S^2$ equates integrals over the sphere to sums over these representations, yielding the expansion of functions in [spherical harmonics](/page/Spherical_harmonics) $Y_{jm}$, where the [character](/page/Character)s $\chi_j$ sum to reproduce zonal functions via [orthogonality](/page/Orthogonality). Similarly, for [character](/page/Character) sums on $SU(2)$, the formula simplifies to [orthogonality](/page/Orthogonality) relations $\sum_g \chi_j(g) \overline{\chi_k(g)} = |G| \delta_{jk}$, inverting lattice-like discretizations. These applications highlight the role in [quantum mechanics](/page/Quantum_mechanics) and [geometry](/page/Geometry), where representation dimensions encode multiplicity in the dual sum.[](https://link.springer.com/article/10.1007/s13324-016-0156-2)