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Trigonometric polynomial
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In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series.

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.

The term trigonometric polynomial for the real-valued case can be seen as using the analogy: the functions sin(nx) and cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of , i.e., Laurent polynomials in under the change of variables .

Definition

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Any function T of the form

with complex-valued coefficients and and at least one of the highest-degree coefficients and non-zero, is called a complex trigonometric polynomial of degree N.[1] The cosine and sine are the even and odd parts of the exponential of an imaginary variable, so the trigonometric polynomial can alternately be written as with complex coefficients ⁠⁠ and for all ⁠⁠ from 1 to ⁠⁠.

If the coefficients and are real for all ⁠⁠, then ⁠⁠ is called a real trigonometric polynomial.[2] When using the exponential form, the complex coefficients satisfy for all .[3]

Properties

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A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of ⁠⁠, or as a function on the unit circle.

Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm;[4] this is a special case of the Stone–Weierstrass theorem. More concretely, for every continuous function ⁠⁠ and every ⁠⁠ there exists a trigonometric polynomial ⁠⁠ such that for all ⁠⁠. Fejér's theorem states that the arithmetic means of the partial sums of the Fourier series of ⁠⁠ converge uniformly to ⁠⁠ provided ⁠⁠ is continuous on the circle; these partial sums can be used to approximate ⁠⁠.

A trigonometric polynomial of degree ⁠⁠ has a maximum of ⁠⁠ roots in a real interval ⁠⁠ unless it is the zero function.[5]

Fejér-Riesz theorem

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See also: Polynomial matrix spectral factorization

The Fejér-Riesz theorem states that every positive real trigonometric polynomial satisfying for all , can be represented as the square of the modulus of another (usually complex) trigonometric polynomial such that:[6] Or, equivalently, every Laurent polynomial with that satisfies for all can be written as: for some polynomial and can be chosen to have no zeroes in the open unit disk .[7][8] The Fejér-Riesz theorem arises naturally in spectral theory and the polynomial factorization is also called the spectral factorization (or Wiener-Hopf factorization) of .[9]

Notes

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References

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See also

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Trigonometric polynomial

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A trigonometric polynomial of degree at most nn is a finite of the constant function 11, the cosine functions cos⁡(kx)\cos(kx), and the sine functions sin⁡(kx)\sin(kx) for k=1,2,…,nk = 1, 2, \dots, n, typically written in the form
Tn(x)=a02+∑k=1n(akcos⁡(kx)+bksin⁡(kx)),T_n(x) = \frac{a_0}{2} + \sum_{k=1}^n \left( a_k \cos(kx) + b_k \sin(kx) \right),
where aka_k and bkb_k are real coefficients. Equivalently, it can be expressed using complex exponentials as
p(x)=∑k=−nnckeikx,p(x) = \sum_{k=-n}^n c_k e^{ikx},
with complex coefficients ckc_k. Trigonometric polynomials are inherently 2π2\pi-periodic and form a under pointwise addition and scalar multiplication, with the set {1,cos⁡(kx),sin⁡(kx)∣k=1,…,n}\{1, \cos(kx), \sin(kx) \mid k=1,\dots,n\} serving as a linearly independent basis on the interval [−π,π][-\pi, \pi] with respect to the constant weight function w(x)=1w(x) = 1. This basis exhibits properties, as
∫−ππcos⁡(kx)cos⁡(mx) dx=πδkm,∫−ππsin⁡(kx)sin⁡(mx) dx=πδkm,∫−ππcos⁡(kx)sin⁡(mx) dx=0\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≥1k, m \geq 1, and similar relations hold involving the constant term, facilitating computations like least-squares approximations. 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 Sn(x)S_n(x) of a function's Fourier series coincide exactly with the best least-squares approximation from the space of degree-nn trigonometric polynomials in the L2[−π,π]L^2[-\pi, \pi] norm. By the Weierstrass approximation theorem, the set of all trigonometric polynomials is dense in the space of continuous 2π2\pi-periodic functions equipped with the uniform norm, meaning any such continuous function can be uniformly approximated arbitrarily closely by a trigonometric polynomial. Notable results include the Fejér-Riesz theorem, which asserts that a non-negative trigonometric polynomial f(θ)=∑k=−nnckeikθf(\theta) = \sum_{k=-n}^n c_k e^{ik\theta} (real-valued on the unit circle) can be factored as f(θ)=∣q(eiθ)∣2f(\theta) = |q(e^{i\theta})|^2, where q(z)q(z) is a of degree at most nn with all roots outside the closed unit disk. This factorization has applications in , , and spectral analysis. Additionally, Bernstein's inequality bounds the of a trigonometric polynomial, stating that if ∥Tn∥∞≤1\|T_n\|_\infty \leq 1 on [−π,π][-\pi, \pi], then ∥Tn′∥∞≤n\|T_n'\|_\infty \leq n.

Definition and Representation

Real-Valued Form

A real-valued trigonometric polynomial of degree at most NN is formally defined as a function T(θ)T(\theta) of the form T(θ)=a02+∑k=1N(akcos⁡(kθ)+bksin⁡(kθ)),T(\theta) = \frac{a_0}{2} + \sum_{k=1}^N (a_k \cos(k\theta) + b_k \sin(k\theta)), where a0,ak,bk∈Ra_0, a_k, b_k \in \mathbb{R} are coefficients. The constant term is conventionally written as a0/2a_0/2 to align with the normalization in Fourier series expansions, though it is sometimes denoted simply as a0a_0. The degree NN of such a is the largest kk for which at least one of aka_k or bkb_k is nonzero; if all coefficients beyond some lower index vanish, the degree is accordingly reduced. This notion of degree parallels that in algebraic , reflecting the highest "frequency" component present in the expression. These arise naturally as the partial sums of for 2π2\pi-periodic functions, providing finite approximations to more general periodic signals. For instance, the function T(θ)=cos⁡(θ)+2sin⁡(2θ)T(\theta) = \cos(\theta) + 2 \sin(2\theta) is a trigonometric of degree 2, with a1=1a_1 = 1, b2=2b_2 = 2, and all other coefficients zero. An alternative representation employs complex exponentials, though the real form is preferred for its direct connection to basis functions.

Complex-Valued Form

A complex trigonometric of degree at most NN is formally defined as a function T(θ)=∑k=−NNckeikθT(\theta) = \sum_{k=-N}^{N} c_k e^{i k \theta}, where the coefficients ckc_k are complex numbers and the degree is NN if cN≠0c_N \neq 0 or c−N≠0c_{-N} \neq 0. This representation leverages the complex exponential basis, which is particularly useful in and due to its connection to . The complex form is equivalent to the real-valued through , eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta. Specifically, the coefficients relate as c0=a0/2c_0 = a_0 / 2, ck=(ak−ibk)/2c_k = (a_k - i b_k)/2 for k>0k > 0, and c−k=(ak+ibk)/2c_{-k} = (a_k + i b_k)/2 for k>0k > 0, where aka_k and bkb_k are the real cosine and sine coefficients, respectively. This allows seamless translation between the two forms, preserving the degree and periodic nature with period 2π2\pi. On the unit circle in the , where z=eiθz = e^{i\theta}, the trigonometric T(θ)T(\theta) corresponds to a Laurent P(z)=∑k=−NNckzkP(z) = \sum_{k=-N}^{N} c_k z^k, evaluated at ∣z∣=1|z| = 1. This mapping highlights the analytic structure, facilitating the study of zeros and factorization within . For instance, consider T(θ)=1+eiθ+e−iθT(\theta) = 1 + e^{i\theta} + e^{-i\theta}, which simplifies to 1+2cos⁡θ1 + 2 \cos \theta using , illustrating the ck=c−k‾c_k = \overline{c_{-k}} that ensures T(θ)T(\theta) is real-valued.

Algebraic Properties

Ring Structure

Trigonometric polynomials constitute a over the real numbers R\mathbb{R} or the complex numbers C\mathbb{C}. For the space TNT_N of real-valued trigonometric polynomials of degree at most NN, and are defined on the circle: if T,S∈TNT, S \in T_N and c∈Rc \in \mathbb{R}, then (T+S)(θ)=T(θ)+S(θ)(T + S)(\theta) = T(\theta) + S(\theta) and (cT)(θ)=c⋅T(θ)(c T)(\theta) = c \cdot T(\theta). This has 2N+12N + 1, with a given by the set {1,cos⁡(kθ),sin⁡(kθ)∣k=1,2,…,N}\{1, \cos(k\theta), \sin(k\theta) \mid k = 1, 2, \dots, N\}, which is linearly independent over R\mathbb{R}. In the complex case, the analogous has basis {eikθ∣k=−N,…,N}\{e^{i k \theta} \mid k = -N, \dots, N\}, also of 2N+12N + 1. The full collection of all trigonometric polynomials (of arbitrary degree) is the union over all NN of the finite-dimensional spaces TNT_N, forming an infinite-dimensional over R\mathbb{R} or C\mathbb{C}. Linear independence of the basis elements follows from their with respect to the inner product ⟨f,g⟩=12π∫02πf(θ)g(θ)‾ dθ\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(θ)=a0+a1cos⁡θ+b1sin⁡θT(\theta) = a_0 + a_1 \cos \theta + b_1 \sin \theta and S(θ)=c0+c1cos⁡θ+d1sin⁡θS(\theta) = c_0 + c_1 \cos \theta + d_1 \sin \theta is (a0+c0)+(a1+c1)cos⁡θ+(b1+d1)sin⁡θ(a_0 + c_0) + (a_1 + c_1) \cos \theta + (b_1 + d_1) \sin \theta, which remains in the degree-1 space unless a1+c1=0a_1 + c_1 = 0 and b1+d1=0b_1 + d_1 = 0, in which case the degree drops. Beyond the structure, trigonometric polynomials form a over R\mathbb{R} or C\mathbb{C}, equipped with a ring multiplication defined : (TS)(θ)=T(θ)S(θ)(T S)(\theta) = T(\theta) S(\theta). This operation is associative and distributive over addition, with the constant 1 serving as the multiplicative identity, confirming the structure. The ring is unital and commutative because multiplication of real- or complex-valued functions inherits these properties from the underlying field.

Multiplication and Degree

Trigonometric polynomials form an under pointwise , where the product of two such polynomials is again a trigonometric . If T(θ)T(\theta) is a trigonometric polynomial of degree MM and S(θ)S(\theta) is one of degree NN, then the product TSTS has degree at most M+NM + N; the degree is exactly M+NM + N provided that the leading terms do not cancel. 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 MM can be written as T(θ)=∑k=−MMckeikθT(\theta) = \sum_{k=-M}^{M} c_k e^{i k \theta}, where c±M≠0c_{\pm M} \neq 0 (or at least one is nonzero, with the degree defined symmetrically). Similarly, S(θ)=∑l=−NNdleilθS(\theta) = \sum_{l=-N}^{N} d_l e^{i l \theta}. The coefficients of the product P(θ)=TS(θ)=∑m=−(M+N)M+NpmeimθP(\theta) = TS(\theta) = \sum_{m=-(M+N)}^{M+N} p_m e^{i m \theta} are obtained via : pm=∑kckdm−k,p_m = \sum_{k} c_k d_{m-k}, where the sum runs over all kk such that both ckc_k and dm−kd_{m-k} are defined, i.e., ∣k∣≤M|k| \leq M and ∣m−k∣≤N|m-k| \leq N. This convolution ensures that PP remains a trigonometric polynomial of degree at most M+NM + N. The leading of the product, corresponding to the term ei(M+N)θe^{i(M+N)\theta}, is cMdNc_M d_N, assuming no higher-degree cancellation occurs in lower terms. If the leading coefficients cMc_M and dNd_N are nonzero, this term dominates, confirming the degree is precisely M+NM + 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⁡θ\cos \theta (degree 1) and cos⁡2θ\cos 2\theta (degree 2). Using the identity, cos⁡θ⋅cos⁡2θ=12[cos⁡(3θ)+cos⁡(θ)],\cos \theta \cdot \cos 2\theta = \frac{1}{2} \left[ \cos(3\theta) + \cos(\theta) \right], the result is a trigonometric of degree 3, with the leading term 12cos⁡3θ\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⁡(kθ)\cos(k\theta) and sin⁡(kθ)\sin(k\theta) for integer k≥0k \geq 0, or equivalently of eikθe^{ik\theta} for integer kk, inherit the periodicity of their constituent terms. Each such basis function satisfies cos⁡(k(θ+2π))=cos⁡(kθ)\cos(k(\theta + 2\pi)) = \cos(k\theta) and sin⁡(k(θ+2π))=sin⁡(kθ)\sin(k(\theta + 2\pi)) = \sin(k\theta), or more generally eik(θ+2π)=eikθe^{ik(\theta + 2\pi)} = e^{ik\theta}, ensuring that any non-constant trigonometric polynomial T(θ)T(\theta) obeys T(θ+2π)=T(θ)T(\theta + 2\pi) = T(\theta) for all real θ\theta. Thus, every non-constant trigonometric polynomial is periodic with a fundamental period that divides 2π2\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 {k1,k2,…,km}\{k_1, k_2, \dots, k_m\}, then the minimal period is 2π/d2\pi / d, where d=gcd⁡(k1,k2,…,km)d = \gcd(k_1, k_2, \dots, k_m) is the of these frequencies. This arises because the signal's periodicity aligns with the least common multiple of the individual periods 2π/∣kj∣2\pi / |k_j|, which equivalently yields a fundamental frequency equal to the GCD of the kjk_j. For instance, consider T(θ)=cos⁡(2θ)+sin⁡(3θ)T(\theta) = \cos(2\theta) + \sin(3\theta); here the frequencies are 2 and 3, with gcd⁡(2,3)=1\gcd(2,3) = 1, so the minimal period is 2π/1=2π2\pi / 1 = 2\pi. When viewed on the unit circle, a trigonometric polynomial T(θ)T(\theta) can be expressed in complex form as T(θ)=∑k=−NNckeikθT(\theta) = \sum_{k=-N}^{N} c_k e^{ik\theta}. Substituting z=eiθz = e^{i\theta} maps this to the evaluation of a Laurent polynomial P(z)=∑k=−NNckzkP(z) = \sum_{k=-N}^{N} c_k z^k on the unit circle ∣z∣=1|z| = 1, highlighting the connection between trigonometric and algebraic structures in this periodic setting.

Zeros and Uniqueness

A non-zero trigonometric polynomial of degree NN has at most 2N2N zeros in any interval of length 2π2\pi, counting multiplicities, unless it is identically zero. This bound arises because trigonometric polynomials are periodic with period 2π2\pi, and the zeros within one period determine the distribution over the entire real line. To establish this result, consider a trigonometric T(θ)=∑k=−NNckeikθT(\theta) = \sum_{k=-N}^{N} c_k e^{ik\theta}. Multiply by eiNθe^{iN\theta} to obtain g(θ)=eiNθT(θ)=∑k=02Nakeikθg(\theta) = e^{iN\theta} T(\theta) = \sum_{k=0}^{2N} a_k e^{ik\theta}, which is an analytic trigonometric of degree 2N2N. The zeros of TT coincide with those of gg, and substituting z=eiθz = e^{i\theta} transforms gg into an algebraic of degree 2N2N in zz, whose number at most 2N2N in the . Thus, gg (and hence TT) has at most 2N2N zeros on the unit , corresponding to θ∈[0,2π)\theta \in [0, 2\pi). This finite zero bound implies uniqueness in trigonometric interpolation. Given 2N+12N+1 distinct points θ0,θ1,…,θ2N\theta_0, \theta_1, \dots, \theta_{2N} in [0,2π)[0, 2\pi) and arbitrary values y0,y1,…,y2Ny_0, y_1, \dots, y_{2N}, there exists a unique trigonometric polynomial of degree at most NN that interpolates these values, i.e., T(θj)=yjT(\theta_j) = y_j for j=0,…,2Nj = 0, \dots, 2N. Uniqueness follows because if two such polynomials T1T_1 and T2T_2 existed, their difference T1−T2T_1 - T_2 would be a non-zero polynomial of degree at most NN with 2N+12N+1 zeros, contradicting the zero bound unless T1−T2≡0T_1 - T_2 \equiv 0. For example, the trigonometric polynomial sin⁡θ\sin \theta, which has degree 1, has exactly two zeros in [0,2π)[0, 2\pi), located at θ=0\theta = 0 and θ=π\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([0,2π])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⁡(kθ)\cos(k\theta) and sin⁡(mθ)\sin(m\theta) expressible via angle addition formulas as linear combinations of other trigonometric terms. This subalgebra contains the constant function 1 and separates points on [0,2π][0, 2\pi], meaning that for any distinct θ1,θ2∈[0,2π)\theta_1, \theta_2 \in [0, 2\pi), there exists a trigonometric polynomial pp such that p(θ1)≠p(θ2)p(\theta_1) \neq p(\theta_2). To verify separation of points, consider the generating set {1,cos⁡(kθ),sin⁡(kθ)∣k≥1}\{1, \cos(k\theta), \sin(k\theta) \mid k \geq 1\}. For distinct θ1\theta_1 and θ2\theta_2, the complex function sin⁡(θ)+icos⁡(θ)=eiθ\sin(\theta) + i \cos(\theta) = e^{i\theta} distinguishes them, as eiθ1≠eiθ2e^{i\theta_1} \neq e^{i\theta_2} implies a difference in either the real part cos⁡(θ)\cos(\theta) or imaginary part sin⁡(θ)\sin(\theta); more elementarily, if sin⁡(θ1)=sin⁡(θ2)\sin(\theta_1) = \sin(\theta_2), then cos⁡(θ1)≠cos⁡(θ2)\cos(\theta_1) \neq \cos(\theta_2), and vice versa. 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([0,2π])C([0, 2\pi]) under the uniform topology, meaning that for any continuous function f:[0,2π]→Rf: [0, 2\pi] \to \mathbb{R} and ε>0\varepsilon > 0, there exists a trigonometric polynomial pp such that ∥f−p∥∞<ε\|f - p\|_\infty < \varepsilon. This density result has profound implications for the approximation of periodic functions: any continuous 2π-periodic function on R\mathbb{R} can be uniformly approximated by trigonometric polynomials of sufficiently high degree, bridging algebraic structure with analytic approximation on the circle. Historically, this trigonometric variant stems from '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.

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 ff, the Cesàro mean is defined as σn(f)(θ)=1n+1∑k=0nsk(f)(θ),\sigma_n(f)(\theta) = \frac{1}{n+1} \sum_{k=0}^n s_k(f)(\theta), where sk(f)(θ)s_k(f)(\theta) denotes the kk-th partial sum of the Fourier series of ff, given by sk(f)(θ)=∑m=−kkf^(m)eimθ,s_k(f)(\theta) = \sum_{m=-k}^k \hat{f}(m) e^{im\theta}, with Fourier coefficients f^(m)=12π∫−ππf(ϕ)e−imϕ dϕ\hat{f}(m) = \frac{1}{2\pi} \int_{-\pi}^\pi f(\phi) e^{-im\phi} \, d\phi. The theorem states that σn(f)(θ)→f(θ)\sigma_n(f)(\theta) \to f(\theta) uniformly on [−π,π][-\pi, \pi] as n→∞n \to \infty. 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 σn(f)(θ)=∫−ππf(ϕ)Kn(θ−ϕ) dϕ2π,\sigma_n(f)(\theta) = \int_{-\pi}^\pi f(\phi) K_n(\theta - \phi) \, \frac{d\phi}{2\pi}, where the Fejér kernel Kn(t)K_n(t) is Kn(t)=1n+1[sin⁡((n+1)t/2)sin⁡(t/2)]2.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 Kn(t)K_n(t) is even, periodic with period 2π2\pi, integrates to 2π2\pi over [−π,π][-\pi, \pi], and serves as an approximate identity, concentrating near zero as nn increases while remaining non-negative everywhere. Each σn(f)\sigma_n(f) is itself a trigonometric polynomial of degree at most nn. The uniform convergence guaranteed by Fejér's theorem implies pointwise convergence for continuous ff, providing a constructive method to approximate such functions by trigonometric polynomials via averaging partial sums. This result is a cornerstone in , 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. A illustrative example is the function f(θ)=θf(\theta) = \theta on [−π,π][-\pi, \pi], extended periodically, which exhibits a jump discontinuity at odd multiples of π\pi. Although ff is not continuous, the Cesàro means σn(f)(θ)\sigma_n(f)(\theta) converge pointwise to f(θ)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.

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(θ)=∑k=−nnckeikθT(\theta) = \sum_{k=-n}^{n} c_k e^{ik\theta} with complex coefficients ckc_k such that T(θ)≥0T(\theta) \geq 0 for all real θ\theta, there exists a trigonometric polynomial Q(θ)=∑k=0ndkeikθQ(\theta) = \sum_{k=0}^{n} d_k e^{ik\theta} with complex coefficients dkd_k satisfying T(θ)=Q(θ)Q(e−iθ)‾T(\theta) = Q(\theta) \overline{Q(e^{-i\theta})}, or equivalently, T(θ)=∣Q(θ)∣2T(\theta) = |Q(\theta)|^2 on the unit circle. In the real-valued case, where T(θ)=a0+∑k=1n(akcos⁡(kθ)+bksin⁡(kθ))T(\theta) = a_0 + \sum_{k=1}^{n} (a_k \cos(k\theta) + b_k \sin(k\theta)) with real coefficients and T(θ)≥0T(\theta) \geq 0 for all θ\theta, the representing polynomial QQ has real coefficients and degree at most nn. 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. The theorem provides a canonical factorization that is analytic inside the unit disk when roots are chosen appropriately, ensuring no zeros of Q(z)Q(z) inside the open unit disk. The proof proceeds by associating to T(θ)T(\theta) the Laurent polynomial P(z)=∑k=−nnckzkP(z) = \sum_{k=-n}^{n} c_k z^k, which is non-negative on the unit circle ∣z∣=1|z| = 1. Since P(z)=z−n∑k=02npkzkP(z) = z^{-n} \sum_{k=0}^{2n} p_k z^k after multiplication by znz^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)Q(z) such that P(z)=Q(z)Q(1/zˉ)‾P(z) = Q(z) \overline{Q(1/\bar{z})} on the circle, with the factorization unique up to a unimodular constant. This spectral factorization leverages the symmetry of roots across the unit circle to ensure the non-negativity condition translates to a perfect square representation. 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.

Relation to Laurent Polynomials

Trigonometric polynomials admit a natural representation in complex exponential form as T(θ)=∑k=−nnckeikθT(\theta) = \sum_{k=-n}^{n} c_k e^{i k \theta}, where the coefficients ckc_k are complex numbers. This form establishes a direct correspondence with Laurent polynomials via the substitution z=eiθz = e^{i \theta} on the unit circle ∣z∣=1|z| = 1, yielding the Laurent polynomial L(z)=∑k=−nnckzkL(z) = \sum_{k=-n}^{n} c_k z^k. Thus, T(θ)=L(eiθ)T(\theta) = L(e^{i \theta}), mapping the trigonometric polynomial to the values of the Laurent polynomial restricted to the unit circle. 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 {eikθ}\{ e^{i k \theta} \} mirrors the monomial basis {zk}\{ z^k \} under the identification. For real-valued trigonometric polynomials expressed in terms of sine and cosine, the isomorphism follows from the relations cos⁡θ=z+z−12\cos \theta = \frac{z + z^{-1}}{2} and sin⁡θ=z−z−12i\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. Analytically, the zeros of T(θ)T(\theta) on the real line correspond precisely to the roots of L(z)L(z) lying on the unit circle. Roots of L(z)L(z) inside the unit disk (∣z∣<1|z| < 1) influence the behavior of T(θ)T(\theta) through analytic continuation inward, while exterior roots (∣z∣>1|z| > 1) affect the outward extension, often via reciprocity relations since L(1/zˉ)L(1/\bar{z}) relates to the . This lifting preserves the multiplicity of zeros and enables the study of zero distribution using tools from . For example, consider the trigonometric T(θ)=2−2cos⁡θT(\theta) = 2 - 2 \cos \theta. In exponential form, T(θ)=2−eiθ−e−iθT(\theta) = 2 - e^{i \theta} - e^{-i \theta}, corresponding to the Laurent L(z)=2−z−z−1L(z) = 2 - z - z^{-1}. Multiplying by zz gives the equivalent equation zL(z)=2z−z2−1=0z L(z) = 2z - z^2 - 1 = 0, or z2−2z+1=0z^2 - 2z + 1 = 0, with a double root at z=1z = 1, reflecting the double of T(θ)T(\theta) at θ=0mod  2π\theta = 0 \mod 2\pi. This bijection has significant implications, permitting the application of complex analysis techniques to trigonometric polynomials. For instance, can be employed to locate roots of L(z)L(z) near the unit circle, thereby determining the number and positions of zeros of T(θ)T(\theta) without direct computation on the real line. Such methods are particularly useful in root location problems and stability analysis in applications like .

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 ff, the NN-th partial sum of its Fourier series is sN(f)(θ)=∑k=−NNf^(k)eikθ,s_N(f)(\theta) = \sum_{k=-N}^N \hat{f}(k) e^{i k \theta}, where f^(k)=12π∫02πf(ϕ)e−ikϕ dϕ\hat{f}(k) = \frac{1}{2\pi} \int_0^{2\pi} f(\phi) e^{-i k \phi} \, d\phi are the Fourier coefficients; this sN(f)s_N(f) is a trigonometric polynomial of degree at most NN that approximates ff. For square-integrable functions f∈L2([0,2π])f \in L^2([0, 2\pi]), the partial sums sN(f)s_N(f) converge to ff in the L2L^2 norm, meaning ∥sN(f)−f∥L2→0\|s_N(f) - f\|_{L^2} \to 0 as N→∞N \to \infty, due to the completeness of the trigonometric system in L2L^2. 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 NN—converge to ff. The partial sums preserve energy through a version of Parseval's identity: 12π∫02π∣sN(f)(θ)∣2 dθ=∑k=−NN∣f^(k)∣2,\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 L2L^2 norm of the approximation by the sum of the squared coefficients up to degree NN. A classic example is the Fourier series approximation of the square wave function, defined as f(θ)=−π/4f(\theta) = -\pi/4 for 0<θ<π0 < \theta < \pi and π/4\pi/4 for π<θ<2π\pi < \theta < 2\pi, extended periodically. The partial sums sN(f)s_N(f) exhibit the Gibbs phenomenon near the discontinuities at θ=0,π\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 NN increases, highlighting limitations of pointwise convergence for discontinuous functions.

In Numerical Methods

Trigonometric interpolation seeks a of degree at most NN that passes through given values of a function at 2N+12N+1 equispaced points on the interval [0,2π)[0, 2\pi). The space of such polynomials has dimension 2N+12N+1, ensuring the existence and uniqueness of the interpolant for any distinct set of points, including equispaced ones. This approach is particularly effective for periodic functions, as it leverages the natural basis of sines and cosines. The (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 (FFT) algorithm, which reduces the from O(M2)O(M^2) to O(Mlog⁡M)O(M \log M) for a signal of length M=2N+1M = 2N+1. For example, applying the FFT to a discrete signal of length M=2N+1M=2N+1 yields the DFT coefficients, and the inverse DFT reconstructs the trigonometric polynomial that exactly interpolates the original data points. In terms of error analysis, the Lebesgue constant for equispaced trigonometric interpolation grows logarithmically with NN, specifically as 4π2ln⁡N+O(1)\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. 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 .

References

  1. https://sites.millersville.edu/rbuchanan/math375/TrigPoly.pdf
  2. https://www.math.pku.edu.cn/teachers/litj/notes/numer_anal/WeierstrassApprox_Pinkus.pdf
  3. https://math.gsu.edu/xye/course/na_handout/slides/approx_handout.pdf
  4. https://lall.stanford.edu/engr207c/lectures/whopf_2008_10_27_01.pdf
  5. https://people.csail.mit.edu/wangliang/Papers/Multi-points_Scheme.pdf
  6. http://www.physics.smu.edu/scalise/P4321sp18/fs.pdf
  7. https://faculty.sites.iastate.edu/jia/files/inline-files/fourier-series.pdf
  8. https://archive.org/download/katznelson-1/Katznelson%5B1%5D_text.pdf
  9. https://courses.grainger.illinois.edu/ece401/sp2017/lecture4fourier.pdf
  10. https://arxiv.org/abs/1909.10345
  11. https://www.math.umd.edu/~petersd/666/amsc666notes003.pdf
  12. https://web.math.utk.edu/~freire/teaching/m467f19/Stone-Weierstrass-Notes.pdf
  13. https://www.math.ubc.ca/~malabika/teaching/ubc/spring19/math321/hw4solutions.pdf
  14. https://www.damtp.cam.ac.uk/user/na/PartIIIat/b02.pdf
  15. https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/eb44436072623f60f8f416875d4b2c11_MIT18_102s21_lec16.pdf
  16. http://susanka.org/MMforQR/Fejer.pdf
  17. https://www.ijsrp.org/research-paper-0621/ijsrp-p11437.pdf
  18. https://eudml.org/doc/149442
  19. https://publikationen.bibliothek.kit.edu/270041275/3868557
  20. https://arxiv.org/pdf/1811.06005.pdf
  21. https://arxiv.org/pdf/2503.03247
  22. https://pakovich.github.io/detri.pdf
  23. https://arxiv.org/pdf/1909.10345
  24. https://www.math.ucdavis.edu/~hunter/book/ch7.pdf
  25. https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/859833f65ee4aec6b0fea35a300d10cd_MIT18_102s21_lec15.pdf
  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. https://www.math.umd.edu/~petersd/666/amsc666notes02.pdf
  28. https://personal.math.vt.edu/embree/math5466/lecture7.pdf
  29. https://encyclopediaofmath.org/wiki/Lebesgue_constants
  30. https://epubs.siam.org/doi/book/10.1137/1.9781611970425
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