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Convergence of Fourier series

The convergence of Fourier series concerns the mathematical conditions under which the infinite series expansion of a periodic function f in terms of sines and cosines—or equivalently, complex exponentials—approximates f itself as more terms are included, with key modes including pointwise convergence at specific points, uniform convergence across intervals, and mean-square (L²) convergence in an integral sense.^[1] For functions f in L^2([-\pi, \pi]), the partial sums S_N(f) of the Fourier series converge to f in the L² norm, meaning \frac{1}{2\pi} \int_{-\pi}^{\pi} |S_N(f)(x) - f(x)|^2 dx \to 0 as N \to \infty, a result established via the orthogonality of the basis \{e^{inx}\} and Parseval's identity, which equates the L² norm of f to the sum of squared Fourier coefficients.^[1] This mean-square convergence holds for any square-integrable periodic function, providing a foundational guarantee of approximation in the Hilbert space setting, though it does not imply pointwise convergence everywhere.^[1] Pointwise convergence is more restrictive: under Dirichlet conditions—where f is piecewise continuous with piecewise continuous derivative on [-\pi, \pi]—the Fourier series converges at every point of continuity to f(x), and at jump discontinuities to the average [f(x^-) + f(x^+)]/2, as per the Riemann localization principle, which ensures local behavior determines convergence at a point.^[2] However, even for continuous functions, pointwise convergence may fail at some points, as counterexamples exist where the series diverges despite f \in C([-\pi, \pi]).^[1] A landmark advance is Carleson's theorem (1966), which asserts that for any f \in L^2([-\pi, \pi]), the Fourier series converges pointwise to f(x) almost everywhere with respect to Lebesgue measure, resolving a long-standing conjecture and extending beyond classical conditions, though the proof relies on advanced techniques involving maximal operators and Hardy spaces.^[3] Uniform convergence requires stronger smoothness, such as f continuous with piecewise continuous derivative, ensuring the series converges uniformly to f by the Weierstrass M-test on the coefficients.^[2] These convergence properties underpin applications in signal processing, partial differential equations, and harmonic analysis, highlighting the balance between the series' representational power and the limitations imposed by the function's regularity.^[1]

Foundations

Preliminaries

The Fourier series of a function f \in L^1([-\pi, \pi]) is defined via its Fourier coefficients \hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx for n \in \mathbb{Z}. These coefficients capture the projection of f onto the complex exponentials e^{inx}, which form an orthonormal basis for L^2([-\pi, \pi]) with respect to the inner product \langle f, g \rangle = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \overline{g(x)} \, dx.^[4] The partial sums of the Fourier series are given by

s_N(f)(x) = \sum_{|n| \leq N} \hat{f}(n) e^{inx},

which can be expressed as the convolution s_N(f)(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) D_N(x - t) \, dt, where the Dirichlet kernel is

D_N(x) = \sum_{|n| \leq N} e^{inx} = \frac{\sin((N + 1/2)x)}{\sin(x/2)}.

This closed-form expression for D_N(x) arises from the summation of a finite geometric series. The kernel D_N is an even, $2\pi-periodic trigonometric polynomial that integrates to $2\pi over [-\pi, \pi], ensuring s_N(1) = 1 for the constant function. The Dirichlet kernel exhibits oscillatory behavior, taking both positive and negative values, which contrasts with approximation kernels that are non-negative. The Lebesgue constant \Lambda_N = \frac{1}{2\pi} \|D_N\|_{L^1} satisfies \Lambda_N \sim \frac{4}{\pi^2} \log N as N \to \infty.^[5] This logarithmic growth indicates that the partial sum operators s_N have operator norms (from L^\infty to L^\infty) bounded by a slowly increasing function, highlighting potential challenges in the convergence of s_N(f) to f. The central question in the study of Fourier series convergence is: under what conditions on f does s_N(f)(x) \to f(x) as N \to \infty for each x \in [-\pi, \pi]?

Magnitude of Fourier Coefficients

The magnitude of Fourier coefficients plays a pivotal role in determining the convergence properties of Fourier series, as slower decay generally implies slower convergence of partial sums. A fundamental result establishing the basic decay behavior is the Riemann-Lebesgue lemma, which asserts that if f \in L^1([-\pi, \pi]), then the Fourier coefficients satisfy \hat{f}(n) \to 0 as |n| \to \infty. This lemma, originally due to Riemann in 1867 and generalized by Lebesgue in 1902, highlights that integrability alone ensures the coefficients vanish at infinity, though without specifying the rate.^[6] For functions with greater regularity, sharper decay estimates can be obtained through integration by parts applied to the coefficient formulas. Specifically, if f is absolutely continuous on [-\pi, \pi] with f' \in L^1([-\pi, \pi]), then integration by parts yields |\hat{f}(n)| \leq \|f'\|_{L^1} / |n| for n \neq 0. More generally, if f belongs to the class C^k of k-times continuously differentiable periodic functions, repeated integration by parts gives |\hat{f}(n)| = O(1/|n|^k) as |n| \to \infty. These bounds demonstrate how increased smoothness accelerates the decay, with each derivative contributing an additional factor of $1/|n|.^[7] Refinements of these estimates apply to Hölder and Zygmund classes, providing precise rates tied to the function's modulus of continuity. For f in the Hölder class C^{0,\alpha} with $0 < \alpha \leq 1, the Bernstein-Zygmund estimates imply |\hat{f}(n)| = O(1/|n|^\alpha). In the Zygmund class, where |f(x+h) + f(x-h) - 2f(x)| = O(|h|), the decay is |\hat{f}(n)| = O(1/|n|), akin to the Lipschitz case but with a weaker continuity condition. These results, developed in the context of trigonometric approximation theory, underscore the interplay between local smoothness and global coefficient behavior.^[8] Illustrative examples highlight these decay patterns. For a step function like the square wave, defined as f(x) = -1 for -\pi < x < 0 and f(x) = 1 for $0 < x < \pi, the sine coefficients decay as b_n = 4/(\pi n) for odd n, exhibiting a slow $1/n rate consistent with the function's discontinuity. In contrast, for smooth functions such as C^\infty periodic functions, the coefficients decay polynomially according to the highest derivative class, while analytic functions achieve exponential decay, |\hat{f}(n)| = O(e^{-c|n|}) for some c > 0. These examples affirm that discontinuities lead to sluggish decay, whereas smoothness promotes rapid attenuation essential for effective series approximation.^[7]

Pointwise and Uniform Convergence

Pointwise Convergence

Pointwise convergence concerns the limit of the partial sums s_n(f)(x) = \sum_{k=-n}^n \hat{f}(k) e^{ikx} of the Fourier series of a $2\pi-periodic function f as n \to \infty at individual points x \in \mathbb{R}. If this limit equals f(x) at every x where f is continuous, or the average of the one-sided limits at discontinuities, the series is said to converge pointwise to f. Early results established sufficient conditions based on the regularity of f, while counterexamples later revealed the limitations of such convergence even for integrable functions. A foundational result is the Dirichlet-Jordan theorem, which guarantees pointwise convergence for functions of bounded variation. Specifically, if f is of bounded variation on [-\pi, \pi], then at every point x of continuity, s_n(f)(x) \to f(x), and at a jump discontinuity, s_n(f)(x) \to \frac{f(x+) + f(x-)}{2}.^[9] This theorem extends Dirichlet's earlier criterion for piecewise smooth functions and highlights how total variation controls the local behavior of the series. The Dini-Dirichlet test provides a more localized condition for convergence at a specific point x. If there exists \delta > 0 such that the Dini integral

\int_0^\delta \frac{|f(x+t) + f(x-t) - 2f(x)|}{t} \, dt < \infty,

then s_n(f)(x) \to f(x).^[9] This criterion captures the smoothness of f near x through the integrability of the symmetric difference quotient, encompassing cases like Lipschitz continuity where the integral converges. Functions in the Hölder class C^{0,\alpha} for \alpha > 0, satisfying |f(x) - f(y)| \leq C |x - y|^\alpha for some constant C, also admit pointwise convergence of their Fourier series to f at every point.^[9] The Hölder condition implies the Dini integral is finite, ensuring the partial sums approximate f locally without requiring global bounded variation. Despite these positive results, pointwise convergence fails in general. In 1873, Paul du Bois-Reymond constructed the first example of a continuous function on [-\pi, \pi] whose Fourier series diverges at a point, disproving earlier conjectures about universal convergence for continuous functions.^[10] More dramatically, Andrey Kolmogorov in 1923 exhibited an explicit integrable function in L^1([-\pi, \pi]) whose Fourier series diverges almost everywhere, showing that even integrability does not suffice for pointwise convergence on a set of full measure.^[11] A major advance came with Carleson's theorem in 1966, which resolved a long-standing conjecture by proving that for every f \in L^2(\mathbb{T}), the Fourier series converges pointwise almost everywhere to f.^[12] This result relies on square-integrability to bound the maximal operator associated with the partial sums, marking the boundary beyond which almost everywhere convergence fails, as Kolmogorov's example illustrates for L^1.

Uniform Convergence

Uniform convergence of the Fourier series of a periodic function f on [-\pi, \pi] requires that the supremum norm of the difference between the partial sums s_N(f) and f tends to zero as N \to \infty, ensuring the series approximates f globally without local discrepancies. A sufficient condition for this is the absolute summability of the Fourier coefficients, \sum_{n=-\infty}^\infty |\hat{f}(n)| < \infty. In this case, the Weierstrass M-test applies directly to the series \sum \hat{f}(n) e^{in x}, as the terms are bounded by the summable sequence M_n = |\hat{f}(n)|, independent of x, yielding uniform convergence on the interval.^[9] Zygmund's theorem provides broader criteria: if f is continuous and \sum |\hat{f}(n)| < \infty, the Fourier series converges uniformly to f. Additionally, for functions in the class C^1, where f is continuously differentiable, the Fourier series also converges uniformly to f, despite the coefficients decaying only as O(1/|n|), which prevents absolute summability. This result follows from integration by parts on the coefficients and properties of the Dirichlet kernel.^[9] The rate of uniform convergence improves with higher smoothness. For f \in C^p with p \geq 1, the error satisfies \|s_N(f) - f\|_\infty = O\left(\frac{\log N}{N^p}\right), reflecting the O(1/N^p) decay of coefficients combined with the O(\log N) growth of the Lebesgue constant for the partial sum operator.^[13] However, uniform convergence does not hold for all continuous functions. Du Bois-Reymond constructed a counterexample in 1873: a continuous periodic function whose Fourier series diverges at a point, implying the partial sums cannot converge uniformly to the function across the interval. Such examples highlight that continuity alone is insufficient for uniform convergence.^[14] Uniform convergence of Fourier series connects to best uniform approximation by trigonometric polynomials via Jackson's theorem, which states that for f \in C^p[-\pi, \pi], the best approximation error E_N(f) by degree-N trigonometric polynomials satisfies E_N(f) \leq C_p \omega_p(f, 1/N), where \omega_p is the modulus of smoothness of order p and C_p is a constant. Since the Fourier partial sum s_N(f) is a trigonometric polynomial of degree N, its approximation error is bounded above by a multiple of the best error plus the Lebesgue constant factor, linking the two concepts in estimating convergence rates.^[15]

Absolute and Norm Convergence

Absolute Convergence

Absolute convergence of the Fourier series of a function f on the circle \mathbb{T} occurs when the series of the absolute values of its Fourier coefficients is finite, that is, \sum_{n=-\infty}^{\infty} |\hat{f}(n)| < \infty.^[16] This condition ensures that the partial sums of the series are uniformly bounded by the total sum of the absolute coefficients, implying uniform convergence of the Fourier series to f by the Weierstrass M-test. The collection of all such functions forms the Wiener algebra A(\mathbb{T}), a Banach algebra under pointwise multiplication with the norm \|f\|_A = \sum_{n=-\infty}^{\infty} |\hat{f}(n)|. This space is closed under multiplication, as the Fourier coefficients of the product fg satisfy |\widehat{fg}(n)| \leq \sum_{k} |\hat{f}(k)| \cdot |\hat{g}(n-k)|, which is bounded by \|f\|_A \|g\|_A, ensuring the product's coefficients are absolutely summable. Functions in A(\mathbb{T}) are continuous, and the uniform convergence allows for a continuous extension across the boundary in analytic settings. In 1914, Sergei Bernstein established that if a $2\pi-periodic function f belongs to the Hölder class C^{0,\alpha} with \alpha > 1/2, then the Fourier series of f converges absolutely.^[17] This theorem highlights the role of smoothness in promoting absolute summability, as the decay of coefficients for Hölder continuous functions with exponent greater than $1/2 is sufficiently rapid, estimated by |\hat{f}(n)| \lesssim |n|^{-\alpha}, leading to the \ell^1 summability.^[17] Trigonometric polynomials, being finite linear combinations of exponentials e^{int}, always possess absolutely convergent Fourier series, since only finitely many coefficients are nonzero. However, absolute convergence does not hold for all continuous functions on \mathbb{T}; there exist continuous functions whose Fourier coefficients decay too slowly for \ell^1 summability, despite uniform convergence in some cases. Sidon sets provide a spectral characterization related to absolute convergence: a subset \Lambda \subset \mathbb{Z} is a Sidon set if every continuous function on \mathbb{T} with Fourier support in \Lambda has an absolutely convergent Fourier series. For such sets, the absolute summability is guaranteed by the sparse nature of \Lambda, which prevents coefficient interactions that would slow decay; examples include lacunary sequences like \Lambda = \{2^k : k \in \mathbb{N}\}. Absolute convergence implies several useful properties, including the preservation of positivity: if f \geq 0 on \mathbb{T}, the uniform limit of its partial sums remains nonnegative. Additionally, membership in A(\mathbb{T}) ensures a continuous extension of f to the closed disk when f arises as a boundary value, leveraging the uniform convergence on the compact set.^[16]

Norm Convergence

Norm convergence refers to the convergence of the partial sums s_N(f) of the Fourier series of a function f to f in the L^p norm on [-\pi, \pi], where \|g\|_{L^p} = \left( \frac{1}{2\pi} \int_{-\pi}^{\pi} |g(x)|^p \, dx \right)^{1/p} for $1 \leq p < \infty and the essential supremum for p = \infty. This type of convergence is particularly well-behaved in Hilbert spaces like L^2, where the exponential functions form an orthonormal basis. The system \{ e^{i n x} / \sqrt{2\pi} \}_{n \in \mathbb{Z}} constitutes a complete orthonormal basis for L^2[-\pi, \pi].^[13] Consequently, the Riesz-Fischer theorem establishes that for any f \in L^2[-\pi, \pi], the partial sums satisfy \| s_N(f) - f \|_{L^2} \to 0 as N \to \infty.^[9] This result, proven independently by Riesz and Fischer in 1907, equates the space of square-summable Fourier coefficients with L^2 functions via isometric isomorphism.^[18] Accompanying this convergence is Parseval's identity, which states that \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2, where \hat{f}(n) are the Fourier coefficients.^[13] In L^2, the rate of convergence can be quantified using Bessel's inequality, which implies \| s_N(f) - f \|_{L^2}^2 = \|f\|_{L^2}^2 - \sum_{|n| \leq N} |\hat{f}(n)|^2 \leq \|f\|_{L^2}^2 - \sum_{|n| \leq N} |\hat{f}(n)|^2, and the right-hand side tends to zero as N \to \infty by completeness of the basis.^[9] This provides an exact measure of the error in terms of the tail of the coefficient series. For $1 < p < \infty, convergence extends to the L^p norm. Trigonometric polynomials are dense in L^p[-\pi, \pi], and the partial sum operators S_N are uniformly bounded on L^p for p > 1, ensuring \| s_N(f) - f \|_{L^p} \to 0 for any f \in L^p.^[13] This boundedness follows from the Marcinkiewicz interpolation theorem applied to the Hilbert space result at p=2 and weak-type estimates at endpoints.^[19] However, no such convergence holds in L^1 or L^\infty. The Dirichlet kernel D_N(x) = \sum_{|k| \leq N} e^{i k x} satisfies \| D_N \|_{L^1} \sim \frac{4}{\pi^2} \log N \to \infty as N \to \infty, implying the operators S_N are not uniformly bounded on L^1.^[9] Thus, there exist functions in L^1[-\pi, \pi] whose Fourier partial sums diverge in the L^1 norm. Similarly, in L^\infty, the lack of uniform boundedness of S_N prevents norm convergence for all continuous functions.^[13]

Almost Everywhere and Summability

Convergence Almost Everywhere

The convergence of Fourier series almost everywhere refers to the property that the partial sums s_N(f)(x) converge to f(x) at almost every point x in the domain, with respect to Lebesgue measure. This notion is central to understanding the behavior of Fourier series for square-integrable functions and their generalizations to other L^p spaces. A landmark result in this area is Carleson's theorem, which states that if f \in L^2([-\pi, \pi]), then the partial sums s_N(f)(x) converge to f(x) almost everywhere as N \to \infty.^[20] The proof relies on establishing the boundedness of the maximal operator \sup_{N} |s_N(f)(x)| on L^2, achieved through estimates involving the Hilbert transform and a square-function argument that controls the growth of the partial sums. This approach leverages the fact that L^2 orthogonality allows for a decomposition into dyadic blocks, where the Hilbert transform bounds the oscillatory components. In 2024, a blueprint for the formalization of Carleson's theorem in the Lean theorem prover was published, breaking down the proof into verifiable steps.^[21] Hunt extended Carleson's result in 1968 to functions in L^p([-\pi, \pi]) for $1 < p < \infty, showing that s_N(f)(x) \to f(x) almost everywhere.^[22] The key innovation is proving that the maximal partial sum operator is bounded on L^p for p > 1, using interpolation between the L^2 case and the weak-type bounds near the endpoints, combined with the Marcinkiewicz interpolation theorem. However, this almost everywhere convergence fails for p = 1. Even stronger, Kolmogorov provided an L^1 function in 1923 whose Fourier series diverges almost everywhere, underscoring the sharp distinction at the endpoint p=1.^[23] Post-2000 developments have focused on quantitative improvements to the boundedness constants for the maximal operator in L^p with p > 1. For instance, Lacey and Thiele's time-frequency analysis framework yielded explicit logarithmic bounds on the operator norm, later refined by subsequent works that provide near-optimal dependence on p close to 1. Despite these advances, the L^1 case remains negative, with no boundedness for the maximal operator and persistent divergence examples.

Summability Methods

Summability methods offer alternative approaches to achieve convergence of Fourier series when the partial sums diverge, by averaging or transforming the partial sums in a controlled manner. These techniques are essential for functions where direct summation fails, such as certain continuous functions on the torus.^[1] A prominent example is Cesàro summability of order 1, which defines the Cesàro means as

\sigma_N(f)(x) = \frac{1}{N+[1](/page/1)} \sum_{k=0}^N s_k(f)(x),

where s_k(f) denotes the k-th partial sum of the Fourier series of f. These means can be equivalently expressed as a convolution with the Fejér kernel:

F_N(x) = \frac{1}{N+[1](/page/1)} \sum_{k=0}^N D_k(x) = \frac{1}{N+[1](/page/1)} \left( \frac{\sin((N+[1](/page/1))x/2)}{\sin(x/2)} \right)^2,

where D_k(x) is the Dirichlet kernel; the Fejér kernel is nonnegative and satisfies \int_{-\pi}^{\pi} F_N(x) \, dx = 2\pi.^[24] The positivity and unit integral property of F_N ensure that the means act as approximation operators that smooth the function while preserving its integral.^[1] Fejér's theorem from 1904 establishes that if f is continuous on [-\pi, \pi], then \sigma_N(f) converges uniformly to f as N \to \infty.^[24] Additionally, for f \in L^1[-\pi, \pi], the Cesàro means converge to f in the L^1 norm.^[1] Higher-order summability methods extend this framework; for instance, Abel summability uses a radial averaging parameter r \uparrow 1 and the Poisson kernel to form means that converge via radial limits in the unit disk. These means recover f uniformly for continuous f and almost everywhere for L^1 functions.^[1] A illustrative example is the sawtooth function f(x) = (\pi - x)/2 for $0 < x < 2\pi, extended periodically, whose Fourier series \sum_{k=1}^\infty \sin(kx)/k diverges at points of discontinuity like x = 0, but the Cesàro means \sigma_N(f) converge to f(x) at every point of continuity and to the average at jumps.^[1] Despite these successes, summability does not guarantee convergence of the original Fourier series, as there are functions where the Cesàro or Abel means converge while the partial sums remain divergent.^[1]

Advanced Topics

Order of Growth

The order of growth of the partial sums of Fourier series is closely tied to the properties of the Dirichlet kernel, which governs the summation process. The Lebesgue constant, defined as the L^1 norm of the Dirichlet kernel D_N, satisfies \|D_N\|_{L^1} \sim \frac{4}{\pi^2} \log N. This logarithmic growth implies that the operator norm of the partial sum projection s_N on C(\mathbb{T}) is also O(\log N), meaning that for any continuous function f, the partial sums satisfy \|s_N(f)\|_\infty \leq C \log N \|f\|_\infty for some constant C > 0. This growth rate manifests in phenomena like the Gibbs overshoot near discontinuities of the function being approximated. For a function with a jump discontinuity, the partial sums exhibit oscillations that do not diminish with increasing N; instead, the overshoot approaches approximately 9% of the jump size as N \to \infty, a consequence of the integral \int_0^\pi \frac{\sin t}{t} \, dt \approx 1.85194, leading to an overshoot of \left( \frac{1}{\pi} \int_0^\pi \frac{\sin t}{t} \, dt - \frac{1}{2} \right) \approx 0.0895 times the jump. Zygmund's theorem provides a precise characterization of this logarithmic growth, establishing that for continuous functions on the circle, the supremum norm of the partial sums grows no faster than O(\log N), with equality achieved for certain examples where the growth is asymptotically \frac{4}{\pi^2} \log N. For smoother functions, such as those in the Lipschitz class, the pointwise maximal operator \sup_N |s_N(f)(x)| is bounded by C \log N \|f\|_{L^\infty}, reflecting controlled overshoots despite the kernel's oscillatory nature. The relatively slow logarithmic growth of these partial sums has profound implications for divergence. Even when Fourier coefficients decay (e.g., slower than $1/n), the \log N factor can amplify oscillations sufficiently to prevent convergence at certain points, enabling counterexamples of divergent series for continuous functions. This was first demonstrated by du Bois-Reymond in 1873, who constructed a continuous function whose Fourier series diverges at a point by exploiting the unbounded growth of the partial sums along a suitable sequence.

Multiple Dimensions

In multiple dimensions, the convergence of Fourier series is studied on the d-dimensional torus \mathbb{T}^d = [0, 2\pi]^d, where functions f: \mathbb{T}^d \to \mathbb{C} are expanded using multi-indexed coefficients \hat{f}(k) = \frac{1}{(2\pi)^d} \int_{\mathbb{T}^d} f(x) e^{-i k \cdot x} \, dx for k \in \mathbb{Z}^d.^[25] The partial sums differ based on the summation region in the frequency domain: square (or rectangular) partial sums \sigma_N f(x) = \sum_{|k_j| \leq N \ \forall j=1,\dots,d} \hat{f}(k) e^{i k \cdot x} aggregate over hypercubes, while circular (or spherical) partial sums \rho_R f(x) = \sum_{|k| \leq R} \hat{f}(k) e^{i k \cdot x} sum over balls of radius R.^[26] For square partial sums, Carleson-Hunt-type theorems establish almost everywhere pointwise convergence for functions in L^p(\mathbb{T}^d) with p > 1, extending the one-dimensional results to higher dimensions via variation norm estimates on maximal operators.^[26] In contrast, circular partial sums present significant challenges: almost everywhere convergence remains an open problem even for L^2(\mathbb{T}^d) functions, while Fefferman constructed a continuous function on \mathbb{T}^2 whose circular partial sums diverge pointwise at some points.^[27] Norm convergence of partial sums to the function holds in L^2(\mathbb{T}^d) for both summation types, as the exponentials e^{i k \cdot x} form a complete orthonormal basis.^[25] For L^p(\mathbb{T}^d) with $1 < p < \infty, p \neq 2, norm convergence occurs for square partial sums but fails for circular partial sums in dimensions d \geq 2, where the operators are not bounded on L^p.^[28]^[27] In special cases, the product structure of the torus \mathbb{T}^d = \mathbb{T} \times \cdots \times \mathbb{T} simplifies analysis for separable functions f(x_1, \dots, x_d) = g_1(x_1) \cdots g_d(x_d), where the multiple Fourier series reduces to the tensor product of one-dimensional series, inheriting their convergence properties.^[28]

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Charles Fefferman "On the convergence of multiple Fourier series," Bulletin of the American Mathematical Society, Bull. Amer. Math. Soc. 77(5), 744-745 ...
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[PDF] Multiple Fourier Series - UNM Math
Nov 30, 2016 · Multiple Fourier series extend Fourier series to higher dimensions, using vector notation. In 2D, it uses complex exponentials, and in dD, it ...