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Trigonometric polynomial

A trigonometric polynomial of degree at most n is a finite linear combination of the constant function $1, the cosine functions \cos(kx), and the sine functions \sin(kx) for k = 1, 2, \dots, n, typically written in the form
T_n(x) = \frac{a_0}{2} + \sum_{k=1}^n \left( a_k \cos(kx) + b_k \sin(kx) \right),
where a_k and b_k are real coefficients.^[1] Equivalently, it can be expressed using complex exponentials as
p(x) = \sum_{k=-n}^n c_k e^{ikx},
with complex coefficients c_k.^[2] Trigonometric polynomials are inherently $2\pi-periodic and form a vector space under pointwise addition and scalar multiplication, with the set \{1, \cos(kx), \sin(kx) \mid k=1,\dots,n\} serving as a linearly independent basis on the interval [-\pi, \pi] with respect to the constant weight function w(x) = 1.^[1] This basis exhibits orthogonality properties, as
\int_{-\pi}^{\pi} \cos(kx) \cos(mx) \, dx = \pi \delta_{km}, \quad \int_{-\pi}^{\pi} \sin(kx) \sin(mx) \, dx = \pi \delta_{km}, \quad \int_{-\pi}^{\pi} \cos(kx) \sin(mx) \, dx = 0
for k, m \geq 1, and similar relations hold involving the constant term, facilitating computations like least-squares approximations.^[3] In approximation theory and harmonic analysis, trigonometric polynomials play a central role as the finite-dimensional building blocks of Fourier series, where the partial sums S_n(x) of a function's Fourier series coincide exactly with the best least-squares approximation from the space of degree-n trigonometric polynomials in the L^2[-\pi, \pi] norm.^[3] By the Weierstrass approximation theorem, the set of all trigonometric polynomials is dense in the space of continuous $2\pi-periodic functions equipped with the uniform norm, meaning any such continuous function can be uniformly approximated arbitrarily closely by a trigonometric polynomial.^[2] Notable results include the Fejér-Riesz theorem, which asserts that a non-negative trigonometric polynomial f(\theta) = \sum_{k=-n}^n c_k e^{ik\theta} (real-valued on the unit circle) can be factored as f(\theta) = |q(e^{i\theta})|^2, where q(z) is a polynomial of degree at most n with all roots outside the closed unit disk.^[4] This factorization has applications in signal processing, control theory, and spectral analysis. Additionally, Bernstein's inequality bounds the derivative of a trigonometric polynomial, stating that if \|T_n\|_\infty \leq 1 on [-\pi, \pi], then \|T_n'\|_\infty \leq n.^[5]

Definition and Representation

Real-Valued Form

A real-valued trigonometric polynomial of degree at most N is formally defined as a function T(\theta) of the form

T(\theta) = \frac{a_0}{2} + \sum_{k=1}^N (a_k \cos(k\theta) + b_k \sin(k\theta)),

where a_0, a_k, b_k \in \mathbb{R} are coefficients.^[6] The constant term is conventionally written as a_0/2 to align with the normalization in Fourier series expansions, though it is sometimes denoted simply as a_0.^[6] The degree N of such a polynomial is the largest integer k for which at least one of a_k or b_k is nonzero; if all coefficients beyond some lower index vanish, the degree is accordingly reduced.^[6] This notion of degree parallels that in algebraic polynomials, reflecting the highest "frequency" component present in the expression. These polynomials arise naturally as the partial sums of Fourier series for $2\pi-periodic functions, providing finite approximations to more general periodic signals.^[6] For instance, the function T(\theta) = \cos(\theta) + 2 \sin(2\theta) is a trigonometric polynomial of degree 2, with a_1 = 1, b_2 = 2, and all other coefficients zero. An alternative representation employs complex exponentials, though the real form is preferred for its direct connection to sine and cosine basis functions.

Complex-Valued Form

A complex trigonometric polynomial of degree at most N is formally defined as a function T(\theta) = \sum_{k=-N}^{N} c_k e^{i k \theta}, where the coefficients c_k are complex numbers and the degree is N if c_N \neq 0 or c_{-N} \neq 0.^[7] This representation leverages the complex exponential basis, which is particularly useful in harmonic analysis and signal processing due to its connection to Fourier series. The complex form is equivalent to the real-valued trigonometric polynomial through Euler's formula, e^{i\theta} = \cos \theta + i \sin \theta. Specifically, the coefficients relate as c_0 = a_0 / 2, c_k = (a_k - i b_k)/2 for k > 0, and c_{-k} = (a_k + i b_k)/2 for k > 0, where a_k and b_k are the real cosine and sine coefficients, respectively.^[7] This bijection allows seamless translation between the two forms, preserving the degree and periodic nature with period $2\pi. On the unit circle in the complex plane, where z = e^{i\theta}, the trigonometric polynomial T(\theta) corresponds to a Laurent polynomial P(z) = \sum_{k=-N}^{N} c_k z^k, evaluated at |z| = 1. This mapping highlights the analytic structure, facilitating the study of zeros and factorization within complex analysis. For instance, consider T(\theta) = 1 + e^{i\theta} + e^{-i\theta}, which simplifies to $1 + 2 \cos \theta using Euler's formula, illustrating the symmetry c_k = \overline{c_{-k}} that ensures T(\theta) is real-valued.^[7]

Algebraic Properties

Ring Structure

Trigonometric polynomials constitute a vector space over the real numbers \mathbb{R} or the complex numbers \mathbb{C}. For the space T_N of real-valued trigonometric polynomials of degree at most N, addition and scalar multiplication are defined pointwise on the circle: if T, S \in T_N and c \in \mathbb{R}, then (T + S)(\theta) = T(\theta) + S(\theta) and (c T)(\theta) = c \cdot T(\theta). This space has dimension $2N + 1, with a standard basis given by the set \{1, \cos(k\theta), \sin(k\theta) \mid k = 1, 2, \dots, N\}, which is linearly independent over \mathbb{R}. In the complex case, the analogous space has basis \{e^{i k \theta} \mid k = -N, \dots, N\}, also of dimension $2N + 1. The full collection of all trigonometric polynomials (of arbitrary degree) is the union over all N of the finite-dimensional spaces T_N, forming an infinite-dimensional vector space over \mathbb{R} or \mathbb{C}. Linear independence of the basis elements follows from their orthogonality with respect to the inner product \langle f, g \rangle = \frac{1}{2\pi} \int_0^{2\pi} f(\theta) \overline{g(\theta)} \, d\theta, where distinct basis functions yield zero inner product. For example, the sum of two degree-1 polynomials T(\theta) = a_0 + a_1 \cos \theta + b_1 \sin \theta and S(\theta) = c_0 + c_1 \cos \theta + d_1 \sin \theta is (a_0 + c_0) + (a_1 + c_1) \cos \theta + (b_1 + d_1) \sin \theta, which remains in the degree-1 space unless a_1 + c_1 = 0 and b_1 + d_1 = 0, in which case the degree drops. Beyond the vector space structure, trigonometric polynomials form a commutative algebra over \mathbb{R} or \mathbb{C}, equipped with a ring multiplication defined pointwise: (T S)(\theta) = T(\theta) S(\theta). This operation is associative and distributive over addition, with the constant polynomial 1 serving as the multiplicative identity, confirming the commutative ring structure. The ring is unital and commutative because pointwise multiplication of real- or complex-valued functions inherits these properties from the underlying field.

Multiplication and Degree

Trigonometric polynomials form an algebra under pointwise multiplication, where the product of two such polynomials is again a trigonometric polynomial. If T(\theta) is a trigonometric polynomial of degree M and S(\theta) is one of degree N, then the product TS has degree at most M + N; the degree is exactly M + N provided that the leading terms do not cancel.^[8] This additive property of degrees mirrors that of ordinary polynomials and follows from the finite support of their Fourier coefficients. In the complex exponential form, a trigonometric polynomial of degree M can be written as T(\theta) = \sum_{k=-M}^{M} c_k e^{i k \theta}, where c_{\pm M} \neq 0 (or at least one is nonzero, with the degree defined symmetrically). Similarly, S(\theta) = \sum_{l=-N}^{N} d_l e^{i l \theta}. The coefficients of the product P(\theta) = TS(\theta) = \sum_{m=-(M+N)}^{M+N} p_m e^{i m \theta} are obtained via convolution:

p_m = \sum_{k} c_k d_{m-k},

where the sum runs over all k such that both c_k and d_{m-k} are defined, i.e., |k| \leq M and |m-k| \leq N. This convolution ensures that P remains a trigonometric polynomial of degree at most M + N.^[8] The leading coefficient of the product, corresponding to the term e^{i(M+N)\theta}, is c_M d_N, assuming no higher-degree cancellation occurs in lower terms. If the leading coefficients c_M and d_N are nonzero, this term dominates, confirming the degree is precisely M + N. In the real-valued form using sines and cosines, the leading behavior is analogous, though the symmetric structure (pairing positive and negative frequencies) may introduce additional terms of the same degree. For a concrete illustration, consider the product of \cos \theta (degree 1) and \cos 2\theta (degree 2). Using the identity,

\cos \theta \cdot \cos 2\theta = \frac{1}{2} \left[ \cos(3\theta) + \cos(\theta) \right],

the result is a trigonometric polynomial of degree 3, with the leading term \frac{1}{2} \cos 3\theta arising from the product of the highest-frequency components in their exponential expansions. This example demonstrates the degree addition without cancellation of the leading term.

Analytic Properties

Periodicity

Trigonometric polynomials, being finite linear combinations of the functions \cos(k\theta) and \sin(k\theta) for integer k \geq 0, or equivalently of e^{ik\theta} for integer k, inherit the periodicity of their constituent terms. Each such basis function satisfies \cos(k(\theta + 2\pi)) = \cos(k\theta) and \sin(k(\theta + 2\pi)) = \sin(k\theta), or more generally e^{ik(\theta + 2\pi)} = e^{ik\theta}, ensuring that any non-constant trigonometric polynomial T(\theta) obeys T(\theta + 2\pi) = T(\theta) for all real \theta. Thus, every non-constant trigonometric polynomial is periodic with a fundamental period that divides $2\pi. The minimal (or fundamental) period of a trigonometric polynomial is determined by the frequencies involved. Specifically, if the nonzero coefficients correspond to the set of integers \{k_1, k_2, \dots, k_m\}, then the minimal period is $2\pi / d, where d = \gcd(k_1, k_2, \dots, k_m) is the greatest common divisor of these frequencies. This arises because the signal's periodicity aligns with the least common multiple of the individual periods $2\pi / |k_j|, which equivalently yields a fundamental frequency equal to the GCD of the k_j. For instance, consider T(\theta) = \cos(2\theta) + \sin(3\theta); here the frequencies are 2 and 3, with \gcd(2,3) = 1, so the minimal period is $2\pi / 1 = 2\pi.^[9] When viewed on the unit circle, a trigonometric polynomial T(\theta) can be expressed in complex form as T(\theta) = \sum_{k=-N}^{N} c_k e^{ik\theta}. Substituting z = e^{i\theta} maps this to the evaluation of a Laurent polynomial P(z) = \sum_{k=-N}^{N} c_k z^k on the unit circle |z| = 1, highlighting the connection between trigonometric and algebraic structures in this periodic setting.^[10]

Zeros and Uniqueness

A non-zero trigonometric polynomial of degree N has at most $2N zeros in any interval of length $2\pi, counting multiplicities, unless it is identically zero.^[11] This bound arises because trigonometric polynomials are periodic with period $2\pi, and the zeros within one period determine the distribution over the entire real line. To establish this result, consider a trigonometric polynomial T(\theta) = \sum_{k=-N}^{N} c_k e^{ik\theta}. Multiply by e^{iN\theta} to obtain g(\theta) = e^{iN\theta} T(\theta) = \sum_{k=0}^{2N} a_k e^{ik\theta}, which is an analytic trigonometric polynomial of degree $2N. The zeros of T coincide with those of g, and substituting z = e^{i\theta} transforms g into an algebraic polynomial of degree $2N in z, whose roots number at most $2N in the complex plane. Thus, g (and hence T) has at most $2N zeros on the unit circle, corresponding to \theta \in [0, 2\pi).^[11] This finite zero bound implies uniqueness in trigonometric interpolation. Given $2N+1 distinct points \theta_0, \theta_1, \dots, \theta_{2N} in [0, 2\pi) and arbitrary values y_0, y_1, \dots, y_{2N}, there exists a unique trigonometric polynomial of degree at most N that interpolates these values, i.e., T(\theta_j) = y_j for j = 0, \dots, 2N. Uniqueness follows because if two such polynomials T_1 and T_2 existed, their difference T_1 - T_2 would be a non-zero polynomial of degree at most N with $2N+1 zeros, contradicting the zero bound unless T_1 - T_2 \equiv 0.^[11] For example, the trigonometric polynomial \sin \theta, which has degree 1, has exactly two zeros in [0, 2\pi), located at \theta = 0 and \theta = \pi. This achieves the maximum number of zeros permitted by the bound.

Approximation and Density

Stone–Weierstrass Application

The Stone–Weierstrass theorem provides a powerful algebraic framework for establishing the density of trigonometric polynomials in the space of continuous functions on the circle. Specifically, the set of all trigonometric polynomials forms a subalgebra of the continuous functions C([0, 2\pi]) equipped with the uniform norm, as it is closed under addition, scalar multiplication, and pointwise multiplication, with products of basis functions like \cos(k\theta) and \sin(m\theta) expressible via angle addition formulas as linear combinations of other trigonometric terms.^[12] This subalgebra contains the constant function 1 and separates points on [0, 2\pi], meaning that for any distinct \theta_1, \theta_2 \in [0, 2\pi), there exists a trigonometric polynomial p such that p(\theta_1) \neq p(\theta_2).^[12]^[13] To verify separation of points, consider the generating set \{1, \cos(k\theta), \sin(k\theta) \mid k \geq 1\}. For distinct \theta_1 and \theta_2, the complex function \sin(\theta) + i \cos(\theta) = e^{i\theta} distinguishes them, as e^{i\theta_1} \neq e^{i\theta_2} implies a difference in either the real part \cos(\theta) or imaginary part \sin(\theta); more elementarily, if \sin(\theta_1) = \sin(\theta_2), then \cos(\theta_1) \neq \cos(\theta_2), and vice versa.^[13]^[12] The subalgebra does not vanish identically at any point, since it includes the nonzero constant 1. By the Stone–Weierstrass theorem, these properties ensure that trigonometric polynomials are dense in C([0, 2\pi]) under the uniform topology, meaning that for any continuous function f: [0, 2\pi] \to \mathbb{R} and \varepsilon > 0, there exists a trigonometric polynomial p such that \|f - p\|_\infty < \varepsilon.^[13]^[12] This density result has profound implications for the approximation of periodic functions: any continuous 2π-periodic function on \mathbb{R} can be uniformly approximated by trigonometric polynomials of sufficiently high degree, bridging algebraic structure with analytic approximation on the circle.^[14] Historically, this trigonometric variant stems from Karl Weierstrass's 1885 theorem, which first demonstrated such density through explicit constructions, later generalized algebraically by Marshall Stone in 1937 to encompass broader settings like compact Hausdorff spaces.^[14]^[2]

Fejér's Theorem

Fejér's theorem addresses the convergence of Cesàro means of Fourier series for continuous periodic functions, establishing that these means, which are trigonometric polynomials, approximate the original function uniformly. Specifically, for a continuous 2π-periodic function f, the Cesàro mean is defined as

\sigma_n(f)(\theta) = \frac{1}{n+1} \sum_{k=0}^n s_k(f)(\theta),

where s_k(f)(\theta) denotes the k-th partial sum of the Fourier series of f, given by

s_k(f)(\theta) = \sum_{m=-k}^k \hat{f}(m) e^{im\theta},

with Fourier coefficients \hat{f}(m) = \frac{1}{2\pi} \int_{-\pi}^\pi f(\phi) e^{-im\phi} \, d\phi. The theorem states that \sigma_n(f)(\theta) \to f(\theta) uniformly on [-\pi, \pi] as n \to \infty.^[15] This uniform convergence can be expressed through convolution with the Fejér kernel, a key non-negative trigonometric polynomial. The Cesàro mean admits the integral representation

\sigma_n(f)(\theta) = \int_{-\pi}^\pi f(\phi) K_n(\theta - \phi) \, \frac{d\phi}{2\pi},

where the Fejér kernel K_n(t) is

K_n(t) = \frac{1}{n+1} \left[ \frac{\sin\left( (n+1) t / 2 \right)}{\sin\left( t / 2 \right)} \right]^2.

The kernel K_n(t) is even, periodic with period $2\pi, integrates to $2\pi over [-\pi, \pi], and serves as an approximate identity, concentrating near zero as n increases while remaining non-negative everywhere. Each \sigma_n(f) is itself a trigonometric polynomial of degree at most n.^[16] The uniform convergence guaranteed by Fejér's theorem implies pointwise convergence for continuous f, providing a constructive method to approximate such functions by trigonometric polynomials via averaging partial sums. This result is a cornerstone in Fourier analysis, originally proved by Lipót Fejér in 1904, and later detailed in standard treatments. As a corollary, it underscores the density of trigonometric polynomials in the continuous functions on the circle, aligning with the Stone–Weierstrass theorem.^[15] A illustrative example is the function f(\theta) = \theta on [-\pi, \pi], extended periodically, which exhibits a jump discontinuity at odd multiples of \pi. Although f is not continuous, the Cesàro means \sigma_n(f)(\theta) converge pointwise to f(\theta) at points of continuity and smooth the jump discontinuities, approaching the average value at the jumps, demonstrating the averaging effect of the Fejér kernel.^[16]

Representation Theorems

Fejér-Riesz Theorem

The Fejér–Riesz theorem asserts that every non-negative trigonometric polynomial can be expressed as the modulus squared of another trigonometric polynomial of the same degree. Specifically, for a trigonometric polynomial T(\theta) = \sum_{k=-n}^{n} c_k e^{ik\theta} with complex coefficients c_k such that T(\theta) \geq 0 for all real \theta, there exists a trigonometric polynomial Q(\theta) = \sum_{k=0}^{n} d_k e^{ik\theta} with complex coefficients d_k satisfying T(\theta) = Q(\theta) \overline{Q(e^{-i\theta})}, or equivalently, T(\theta) = |Q(\theta)|^2 on the unit circle.^[17] In the real-valued case, where T(\theta) = a_0 + \sum_{k=1}^{n} (a_k \cos(k\theta) + b_k \sin(k\theta)) with real coefficients and T(\theta) \geq 0 for all \theta, the representing polynomial Q has real coefficients and degree at most n.^[18] This result was proved by Frigyes Riesz around 1911 and independently published by Lipót Fejér in 1915, who attributed the proof to Riesz in his paper.^[19] The theorem provides a canonical factorization that is analytic inside the unit disk when roots are chosen appropriately, ensuring no zeros of Q(z) inside the open unit disk.^[17] The proof proceeds by associating to T(\theta) the Laurent polynomial P(z) = \sum_{k=-n}^{n} c_k z^k, which is non-negative on the unit circle |z| = 1. Since P(z) = z^{-n} \sum_{k=0}^{2n} p_k z^k after multiplication by z^n, the roots of the resulting polynomial come in pairs symmetric with respect to the unit circle (reciprocals). By selecting for each pair the root outside or on the unit circle and forming the product, one constructs Q(z) such that P(z) = Q(z) \overline{Q(1/\bar{z})} on the circle, with the factorization unique up to a unimodular constant.^[17] This spectral factorization leverages the symmetry of roots across the unit circle to ensure the non-negativity condition translates to a perfect square representation.^[20] The theorem has significant applications in the solution of moment problems on the unit circle, where non-negative trigonometric polynomials arise as moment sequences, and in the theory of orthogonal polynomials on the circle, facilitating explicit constructions via factorization.^[17]

Relation to Laurent Polynomials

Trigonometric polynomials admit a natural representation in complex exponential form as T(\theta) = \sum_{k=-n}^{n} c_k e^{i k \theta}, where the coefficients c_k are complex numbers. This form establishes a direct correspondence with Laurent polynomials via the substitution z = e^{i \theta} on the unit circle |z| = 1, yielding the Laurent polynomial L(z) = \sum_{k=-n}^{n} c_k z^k. Thus, T(\theta) = L(e^{i \theta}), mapping the trigonometric polynomial to the values of the Laurent polynomial restricted to the unit circle.^[21] Algebraically, this correspondence induces an isomorphism between the ring of trigonometric polynomials and the ring of Laurent polynomials in one variable over the complex numbers. Specifically, the map extends to a ring isomorphism by preserving addition and multiplication, as the exponential basis \{ e^{i k \theta} \} mirrors the monomial basis \{ z^k \} under the identification. For real-valued trigonometric polynomials expressed in terms of sine and cosine, the isomorphism follows from the relations \cos \theta = \frac{z + z^{-1}}{2} and \sin \theta = \frac{z - z^{-1}}{2i}, which generate the Laurent polynomial ring. This structure allows trigonometric identities to be analyzed through algebraic properties of Laurent polynomials.^[22]^[21] Analytically, the zeros of T(\theta) on the real line correspond precisely to the roots of L(z) lying on the unit circle. Roots of L(z) inside the unit disk (|z| < 1) influence the behavior of T(\theta) through analytic continuation inward, while exterior roots (|z| > 1) affect the outward extension, often via reciprocity relations since L(1/\bar{z}) relates to the reciprocal polynomial. This lifting preserves the multiplicity of zeros and enables the study of zero distribution using tools from complex analysis.^[23] For example, consider the trigonometric polynomial T(\theta) = 2 - 2 \cos \theta. In exponential form, T(\theta) = 2 - e^{i \theta} - e^{-i \theta}, corresponding to the Laurent polynomial L(z) = 2 - z - z^{-1}. Multiplying by z gives the equivalent polynomial equation z L(z) = 2z - z^2 - 1 = 0, or z^2 - 2z + 1 = 0, with a double root at z = 1, reflecting the double zero of T(\theta) at \theta = 0 \mod 2\pi.^[21] This bijection has significant implications, permitting the application of complex analysis techniques to trigonometric polynomials. For instance, Rouché's theorem can be employed to locate roots of L(z) near the unit circle, thereby determining the number and positions of zeros of T(\theta) without direct computation on the real line. Such methods are particularly useful in root location problems and stability analysis in applications like signal processing.^[22]

Applications

In Fourier Analysis

In Fourier analysis, trigonometric polynomials arise naturally as the partial sums of Fourier series expansions for periodic functions. For a 2π-periodic function f, the N-th partial sum of its Fourier series is

s_N(f)(\theta) = \sum_{k=-N}^N \hat{f}(k) e^{i k \theta},

where \hat{f}(k) = \frac{1}{2\pi} \int_0^{2\pi} f(\phi) e^{-i k \phi} \, d\phi are the Fourier coefficients; this s_N(f) is a trigonometric polynomial of degree at most N that approximates f. For square-integrable functions f \in L^2([0, 2\pi]), the partial sums s_N(f) converge to f in the L^2 norm, meaning \|s_N(f) - f\|_{L^2} \to 0 as N \to \infty, due to the completeness of the trigonometric system in L^2.^[24] For continuous functions, uniform convergence of the partial sums does not hold in general, but extensions of Fejér's theorem guarantee that the Cesàro means of the partial sums—which are also trigonometric polynomials of degree N—converge uniformly to f.^[15] The partial sums preserve energy through a version of Parseval's identity:

\frac{1}{2\pi} \int_0^{2\pi} |s_N(f)(\theta)|^2 \, d\theta = \sum_{k=-N}^N |\hat{f}(k)|^2,

which bounds the L^2 norm of the approximation by the sum of the squared coefficients up to degree N.^[25] A classic example is the Fourier series approximation of the square wave function, defined as f(\theta) = -\pi/4 for $0 < \theta < \pi and \pi/4 for \pi < \theta < 2\pi, extended periodically. The partial sums s_N(f) exhibit the Gibbs phenomenon near the discontinuities at \theta = 0, \pi, with overshoots that approach approximately 8.95% of the jump height (about 0.089 times the discontinuity size) and do not diminish as N increases, highlighting limitations of pointwise convergence for discontinuous functions.^[26]

In Numerical Methods

Trigonometric interpolation seeks a trigonometric polynomial of degree at most N that passes through given values of a function at $2N+1 equispaced points on the interval [0, 2\pi). The space of such polynomials has dimension $2N+1, ensuring the existence and uniqueness of the interpolant for any distinct set of points, including equispaced ones.^[27] This approach is particularly effective for periodic functions, as it leverages the natural basis of sines and cosines. The discrete Fourier transform (DFT) provides the explicit form of this interpolant, where the coefficients of the trigonometric polynomial are the DFT values scaled appropriately. These coefficients can be computed efficiently using the fast Fourier transform (FFT) algorithm, which reduces the computational complexity from O(M^2) to O(M \log M) for a signal of length M = 2N+1. For example, applying the FFT to a discrete signal of length M=2N+1 yields the DFT coefficients, and the inverse DFT reconstructs the trigonometric polynomial that exactly interpolates the original data points.^[28] In terms of error analysis, the Lebesgue constant for equispaced trigonometric interpolation grows logarithmically with N, specifically as \frac{4}{\pi^2} \ln N + O(1), which contrasts sharply with the exponential growth observed in algebraic polynomial interpolation at equispaced points. This logarithmic growth ensures stable and well-conditioned interpolation for smooth periodic functions.^[29] Trigonometric polynomials play a central role in modern spectral methods for solving partial differential equations (PDEs) on periodic domains, where they form the Fourier basis for high-order spatial discretizations, enabling exponential convergence for smooth solutions since the 1980s.^[30]

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[19]
[PDF] The Riesz-Fejer Theorem: Missing Link between
The Superposition principle is found to be a consequence of an apparently little-known mathematical theorem for non-negative Fourier polynomials published by ...
[20]
https://arxiv.org/pdf/1811.06005.pdf
[21]
https://arxiv.org/pdf/2503.03247
[22]
[PDF] ON DECOMPOSITIONS OF TRIGONOMETRIC POLYNOMIALS
Let P be a polynomial with complex coefficients. Any representation of P in the form P = P1 ◦ W1, where P1 and W1 are polynomials of degree greater than ...
[23]
https://arxiv.org/pdf/1909.10345
[24]
[PDF] Chapter 7: Fourier Series - UC Davis Mathematics
Functions in V are called trigonometric polynomials. We will prove that any continuous function on. Τ can be approximated uniformly by trigonometric polynomials ...
[25]
[PDF] 18.102 S2021 Lecture 15. Orthonormal Bases and Fourier Series
Apr 13, 2021 · Theorem 164 (Parseval's identity). Let H be a Hilbert space, and let ... and the Nth partial Fourier sum is. SNf (x) = X. |n|≤N f(n)einx ...
[26]
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/06%3A_Continuous_Time_Fourier_Series_(CTFS)/6.07%3A_Gibbs_Phenomena
[27]
[PDF] 1 Polynomial approximation and interpolation - 1.1 Jackson theorems
The problem of approximating functions on intervals by polynomials is closely related to the problem of approximating periodic functions by trigonometric ...
[28]
[PDF] lecture 7: Trigonometric Interpolation - Virginia Tech
The coefficient 7` that premultiplies ei`x in the trigonometric interpolating polynomial is actually an approximation of the Fourier coefficient c`. Let us go ...
[29]
[PDF] On the Lebesgue constant of the trigonometric Floater-Hormann ...
Feb 18, 2019 · ... growth of the Lebesgue constant on equally spaced points. We show that the growth is logarithmic providing a stable interpolation operator.
[30]
Numerical Analysis of Spectral Methods | SIAM Publications Library
The focus is on the development of a new mathematical theory that explains why and how well spectral methods work. Included are interesting extensions of the ...