Fourier series

For a Diophantine frequency vector, the equation $\omega \cdot {\partial }_{x}f=g$ on a periodic torus has an analytic solution on each smaller complex strip exactly when $g$ has zero average. Its nonzero Fourier coefficients are $f_k=g_k/(ik\cdot \omega )$, and its zero coefficient is arbitrary. Thus the normalized zero-mean solution is unique.

For a sufficiently small analytic periodic vector field $w$, a near-identity angular diffeomorphism $\chi$ and a constant vector $a$ solve $D\chi \omega =\omega +w\circ \chi +a$. They satisfy $\chi -\mathrm{id}=O(w)$, $a=O(w)$; if $⟨w⟩=0$, then $a=O(w^2)$. For the defect $e={\mathcal{D}}_{\omega }\chi -\omega -w\circ \chi -a$, set $A=D\chi$, choose $\Delta a=⟨A^{-1}{⟩}^{-1}⟨A^{-1}e⟩$, and solve ${\mathcal{D}}_{\omega }v=A^{-1}(\Delta a-e)$ with zero mean. Updating $\chi$ by $Av$ leaves a quadratic defect $(De)v-[w(\chi +Av)-w(\chi )-Dw(\chi )Av]$. The torus small-divisor estimate and geometrically decreasing strip losses yield a convergent analytic iteration.

If $g$ is analytic on a complex strip of width $\sigma$ and has zero mean, the normalized solution satisfies $\Vert f{\Vert }_{\sigma -\delta }\le C_{n,\tau }{\gamma }^{-1}{\delta }^{-(n+\tau )}\Vert g{\Vert }_{\sigma }$. Fourier decay supplies $e^{-\delta |k{|}_1}$ after the strip loss, and lattice-shell counting bounds the remaining sum ${\sum }_{k\ne 0}|k{|}_1^{\tau }{e}^{-\delta |k{|}_1}$. This simple bound is sufficient for analytic conjugacy iterations even when sharper estimates are available.

Every null subset of the circle lies in the divergence set of the Fourier series of some continuous complex-valued function. One can arrange unbounded Fourier partial sums there. The compact-set Fourier amplification lemma, compact batching of a small open set and frequency-separated Fourier block series construct the function by a summable series of small blocks. The result covers nonclosed and dense null sets; it does not require the divergence set to equal the originally specified set.

For an integer $m$, direct integration gives zero unless $m=0$, in which case it gives one. Products of this identity on the unit cube turn integrals of finite exponential sums into counts of integer solutions. This is the elementary orthogonality used in the Vinogradov mean value.

The Fourier series of a periodic function of bounded variation converges at each point to the mean of its two one-sided limits. In particular, it converges to the function at continuity points. Periodizing a compact interval introduces jumps at its integer endpoints; their half-values explain the half-weight convention in the Van der Corput sum-integral lemma. The conclusion is pointwise convergence, not a claim of absolute convergence of every such Fourier series.

The period-one extension of $x(1-x)$ on $[0,1]$ has Fourier series $1/6-{\pi }^{-2}{\sum }_{n\ge 1}{n}^{-2}\cos (2\pi nx)$. Endpoint agreement makes the periodic function continuous. Its derivative series is $(2/\pi ){\sum }_{n\ge 1}{n}^{-1}\sin (2\pi nx)$, which equals $1-2x$ in the open unit interval and converges to zero at integers, the midpoint of the derivative's jump. Termwise differentiation of a Fourier series applies locally away from the corners.

For a periodic function that is piecewise continuously differentiable with finitely many pieces, its Fourier series converges at each point to the average of the two one-sided limits. At a point of continuity this is the function value. This includes the identified interval endpoints, whose two limits belong to the periodic extension. Absolute summability of the Fourier coefficients additionally gives uniform convergence.

The complex form of a Fourier series for period $2L$ uses coefficients $c_n=(2L{)}^{-1}{\int }_{-L}^{L}f(x)e^{-in\pi x/L}dx$ and complex exponentials $e^{in\pi x/L}$. For square-integrable functions it converges in the $L^2$ norm, and the Parseval identity is $(2L{)}^{-1}\int |f{|}^2={\sum }_n|c_n{|}^2$. Real functions satisfy $c_{-n}=\overline{c_n}$.

The absolutely convergent sine series $S(\xi )={\sum }_{l\ge 1}\sin (l\xi )/l^2$ has the integral representation $S(\xi )=\sin \xi {\int }_0^{\infty }t{e}^{-t}/(1-2e^{-t}\cos \xi +e^{-2t})dt$. The integrand is positive for $0<\xi <\pi$, proving $S>0$ there without incorrectly maximizing individual Fourier terms.

The Wiener algebra consists of periodic functions with absolutely summable Fourier coefficients, with norm ${\sum }_k|a_k|$. Multiplication corresponds to discrete convolution of coefficients, so the norm is submultiplicative. Consequently $e^f={\sum }_{n\ge 0}{f}^n/n!$ also belongs to the algebra when $f$ does. Every such Fourier series is uniformly convergent and defines a continuous function.

A sinusoidal component at an integer multiple of a fundamental angular frequency. For a $2\pi$-periodic angular coordinate $\varphi$, real Fourier series use $\cos (k\varphi )$ and $\sin (k\varphi )$. Summing over $q$ equally spaced phases removes every component whose integer $k$ is not a multiple of $q$, since ${\sum }_{j=0}^{q-1}{e}^{2\pi ikj/q}=0$ in that case. Indeed, the geometric series has ratio $w=e^{2\pi ik/q}\ne 1$ and sum $(1-w^q)/(1-w)=0$. When $q$ divides $k$, all $q$ terms are one. This cancellation isolates higher-order feedback in resonant conjunction geometry.

For an integer $k\ge 2$ and $\mathrm{Im}z>0$, differentiating the partial-fraction expansion of $\pi \cot \pi z$ givesBoth differentiated series converge locally uniformly. This kernel computes the Fourier expansion of a modular form for holomorphic Eisenstein series.

For $w\notin \mathbb{Z}$, integrate $\pi \cot (\pi \zeta )/(\zeta -w{)}^2$ around large squares whose sides have half-integer coordinates. The cotangent is uniformly bounded on these sides and the integral tends to zero. The residue theorem gives residues $(n-w{)}^{-2}$ at the integers and $-{\pi }^2{\csc }^2(\pi w)$ at $w$, proving the identity. For $\mathrm{Im}w>0$, the geometric-series formula for the cotangent also gives ${\pi }^2{\csc }^2(\pi w)=-4{\pi }^2{\sum }_{r\ge 1}r{e}^{2\pi irw}$.

On the circle of circumference $2\pi$, the functions $e_k(x)=e^{ikx}$ for $k\in \mathbb{Z}$ form an orthogonal basis of $L^2$.

A Fourier sine series expands a function on an interval in the eigenfunctions $\sin (n\pi x/L)$ of the Dirichlet Laplacian. It is the odd Fourier-series extension across the interval endpoints.

For $0<t\le \pi$, split the sum at $\lfloor 1/t\rfloor$. The first part is bounded using $|\sin kt|\le kt$; summation by parts and the geometric bound on interval sine sums control the rest. The harmonic coefficient sum grows like $\log m$, but cancellations keep the whole trigonometric polynomial uniformly bounded. Modulating this polynomial lets a Fourier partial sum expose the large uncancelled half.

The functions $\sqrt{2/\pi }\sin (nx)$, $n\ge 1$, form a complete orthonormal basis of $L^2(0,\pi )$. An odd extension to $(-\pi ,\pi )$ reduces completeness to that of the Fourier series. Equality of all sine coefficients therefore implies equality almost everywhere; if both functions are continuous, it implies equality everywhere.

The Leibniz formula is the conditionally convergent alternating series

A trigonometric polynomial is a finite linear combination of complex exponentials $e^{inx}$, or equivalently of sines and cosines. If it vanishes on an interval, all its coefficients vanish.

Under the displayed diagonal-sum convention, set $w(z)=(1,z^{-1},\dots ,z^{-d}{)}^T$ on the unit circle. A Hermitian matrix $M$ then givesConsequently $M⪰0$ proves nonnegativity. Conversely the Fejér–Riesz theorem produces the rank-one spectral-factor Gram matrix. This gives a semidefinite programming representation of nonnegative trigonometric polynomials. The reversed convention $j-i=k$ instead uses the unconjugated monomial vector.

For coefficient column $q=(q_0,\dots ,q_d{)}^T$, $M_{ij}=q_i\overline{q_j}$ is a Hermitian matrix and a positive semidefinite matrix because $x^{\ast }Mx=|q^{\ast }x{|}^2$. Expanding $|q(z){|}^2$ on the unit circle gives coefficient $p_k={\sum }_{i-j=k}{M}_{ij}$. Thus $M$ has matrix rank one when $q\ne 0$ and zero otherwise. General feasible Gram matrices need not have matrix rank one.

A trigonometric polynomial nonnegative on the unit circle has a polynomial modulus-square factor of polynomial degree at most its trigonometric order. For a nonzero trigonometric polynomial of actual order $d$, conjugate symmetry of trigonometric polynomial coefficients yields reciprocal-conjugate root pairing for $P=z^{d}p$. Even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial permits pairing all $2d$ roots of a polynomial. Select representatives ${\zeta }_1,\dots ,{\zeta }_d$ and useIt follows that $p(z)=c{\prod }_i|z-{\zeta }_i{|}^2$ with $c=(-1{)}^d{p}_d/{\prod }_i\overline{{\zeta }_i}$. Evaluating away from the roots of a polynomial gives $c>0$. Thus $q(z)=\sqrt{c}{\prod }_i(z-{\zeta }_i)$ works. Constant and zero trigonometric polynomials have constant or zero factors. Different selections of roots of a polynomial and constant phases can give different factors.

If $P=z^{d}p$ and $p$ is nonnegative on the unit circle, a root of a polynomial $\zeta =e^{i{\theta }_0}$ there has even multiplicity of a root. Indeed $p(e^{i\theta })$ is a real analytic nonnegative function, whose first nonzero term in its Taylor series at a zero must have even order. The map $\theta \mapsto e^{i\theta }-\zeta$ has a simple zero, and multiplication by $e^{-id\theta }$ is nonvanishing, so the order equals the polynomial multiplicity.

A trigonometric polynomial is real on the unit circle precisely when its finite coefficients have the displayed symmetry. To prove necessity, conjugate its values on the circle, use $\overline{z}=z^{-1}$, and compare coefficients. Multiplying a coefficient difference by $z^d$ gives an ordinary polynomial vanishing at infinitely many points, so it is zero. Symmetry also gives $p(z^{-1})=\overline{p(\overline{z})}$ for every nonzero complex $z$.

For a nonzero real-on-the-circle trigonometric polynomial of actual order $d$, the polynomial $P(z)=z^{d}p(z)$ satisfies $P(z)=z^{2d}\overline{P(1/\overline{z})}$. Its constant term is the nonzero conjugate of its leading coefficient. Hence its nonzero roots of a polynomial occur in pairs $\zeta ,1/\overline{\zeta }$ with the same multiplicity of a root. The coefficient reversal and complex conjugation preserve the exponents of the paired factors. Fixed roots of a polynomial on the unit circle require nonnegativity, rather than this symmetry alone, to have even multiplicity.

A Fourier partial sum retains the modes with $|k|\le n$.

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.

A continuous complex-valued function can have uniformly bounded Fourier partial sums which fail to converge at one point. Normalize harmonic sine polynomials by their harmonic coefficient sums, modulate them into disjoint high-frequency intervals and choose their degrees so that the complete norms are summable. At zero, complete blocks vanish while their midpoint prefixes have a common nonzero value. The frequency-separated Fourier block series gives a uniform bound on every partial sum as well as two distinct subsequential values.

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

The Fejér kernel is the arithmetic mean of the Dirichlet kernels. It is nonnegative and has normalized integral one.

The $n$th Fejér sum is the arithmetic mean of the first $n$ Fourier partial sums. Equivalently, it is the convolution of the function with the Fejér kernel.

The Fejér kernel is nonnegative and has unit mass in its convolution normalization. The integral triangle inequality therefore bounds the supremum norm of a Fejér sum by that of the original function. In the half-kernel convention $F_n=(2n{)}^{-1}|{\sum }_{j=0}^{n-1}{e}^{ijt}{|}^2$, its integral is $\pi$ and the convolution prefactor is $1/\pi$.

For a function of period $L$, its complex Fourier coefficients are

For a periodic absolutely continuous function with derivative in $L^2$, integration by parts gives $\widehat{f^{\prime }}(n)=in\hat{f}(n)$. Bessel's inequality and the Cauchy-Schwarz inequality prove the displayed bound in the normalized circle norm. Adding the constant coefficient gives an absolutely and uniformly convergent Fourier series. Merely bounding individual coefficients by $O(1/|n|)$ would not establish absolute convergence.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.

If a periodic function extends analytically and remains bounded in a nonzero complex strip around the real axis, contour shifting gives constants $C,c>0$ such thatThis exponential decay underlies spectral convergence of Fourier approximation and periodic trapezoidal quadrature.

When regularity and convergence justify termwise differentiation, the derivative of a Fourier mode multiplies its coefficient by its frequency. For period $L$,

A Fourier cosine series has the formIt represents the even function obtained by reflecting its data across the origin.

On $(-\pi ,\pi )$,Parseval's identity and direct integration of the square give