Fourier series ($f(x)\sim a_0/2+{\sum }_{n\ge 1}[a_n\cos (nx)+b_n\sin (nx)]$)

A Fourier series describes a periodic function through sines and cosines whose frequencies are integer multiples of one base frequency. These Fourier harmonics give a useful alternative to describing the value at every input: each Fourier coefficient measures how much of one oscillating pattern is present. For a real, integrable function of period $2\pi$, the real form isThe cosine formula includes $n=0$; the sine formula uses $n\ge 1$. Thus $a_0/2$ is the mean value over a period, not $a_0$ itself. The symbol $\sim$ associates a function with its Fourier coefficients; an equality requires a specified kind of convergence, such as pointwise convergence. For general period $T>0$, replace $nx$ by $2\pi nt/T$ and the coefficient prefactor by $2/T$. Fourier orthogonality explains why integrating against one mode removes the others. ^[ref].

For a finite example, takeIts mean is $1/2$, its real Fourier coefficients are $a_0=1$, $a_1=1$, $b_2=1/2$, and all other $a_n,b_n$ vanish. At $x=\pi /4$, the three contributions add to $1/2+1/\sqrt{2}+1/2=1+1/\sqrt{2}\approx 1.70710678$. The function is already a trigonometric polynomial, so retaining these modes reconstructs it exactly. Here $x$ is an angular coordinate in radians; if it instead represents time, a base angular frequency must supply the corresponding units.

The change of viewpoint is useful because many linear problems act separately on these modes. The second derivative maps $\sin (nx)$ to $-n^2\sin (nx)$ and maps $\cos (nx)$ in the same way. Consequently a differential equation that is complicated for a whole profile can become a simpler calculation for each Fourier harmonic. A linear time-invariant system similarly has a separate response to each complex exponential. Superposition combines the responses when the relevant sums and operations are justified. ^[ref].

The infinite case contains a subtlety absent from the finite example. Under pointwise convergence of a piecewise smooth Fourier series, the value at a jump discontinuity is the average of the two one-sided limits. For a square-integrable function, the complex Fourier series instead converges in the $L^2$ norm: the integrated squared error tends to zero. Neither statement says that the error becomes small everywhere at once. A square wave shows the difference vividly through the Gibbs phenomenon.

A historical source for this connection between profiles and modes is Joseph Fourier's 1822 book Théorie analytique de la chaleur. Chapter III develops expansions in sines and cosines while studying the propagation of heat. The heat equation gives a continuing application of this approach. ^[ref].

For an integrable periodic function with one-sided limits $L_{+},L_{-}$ at $t$, the displayed condition implies that its classical Fourier partial sums converge to $(L_{+}+L_{-})/2$. To prove it use the Dirichlet kernel $D_n(s)=\sin ((n+1/2)s)/\sin (s/2)$ and its integral $2\pi$. Subtract the target value and combine the positive and negative halves of the convolution integral. The resulting difference isThe coefficient of the sine is integrable near zero by the Dini condition, and away from zero by integrability of $f$. The Riemann-Lebesgue lemma therefore makes this expression tend to zero. Step functions satisfy the condition because the numerator vanishes in a sufficiently small punctured neighborhood. This is a local Fourier convergence test, distinct from Dini's theorem on uniform convergence of monotone sequences of continuous functions.

A Hadamard-lacunary Fourier series has nonzero frequencies supported on a positive sequence with the displayed gap ratio, possibly together with its negatives and finitely many extra modes. Such gaps strongly restrict additive frequency relations. They make moment estimates resemble those of independent random variables. For a ratio greater than three, the strongly lacunary polynomial L1-L2 bound follows directly by enumerating fourth-moment pairings.

Use normalized measure on the circle. If a trigonometric polynomial has frequency support $\{\pm n_j\}$ with $n_{j+1}>3n_j$, every zero sum of four signed frequencies pairs into opposite equal frequencies: the largest unpaired term would exceed the sum of the other three. Expanding the fourth moment and bounding its three possible pairings gives $\Vert p{\Vert }_4^4\le 3\Vert p{\Vert }_2^4$. The Holder inequality gives $\Vert p{\Vert }_2\le \Vert p{\Vert }_1^{1/3}\Vert p{\Vert }_4^{2/3}$ and hence the displayed estimate. Therefore an $L^1$ function with only these lacunary frequencies has square-summable coefficients: apply the estimate to its uniformly $L^1$-bounded Fejér sums and then pass to finite coefficient blocks. Finitely many extra modes do not affect the conclusion.

For positive integers $m,n$, the sine functions form an orthogonal family on $(0,L)$:The same formula holds for the corresponding cosine functions when both indices are positive. Product-to-sum identities reduce the distinct-index products to cosines with nonzero integer frequencies, each with zero integral; for matching indices, the squared sine or cosine has mean $1/2$. This proves the formula and explains why separate normal modes of a uniform fixed-end string make additive contributions to its kinetic energy and potential energy. The constant cosine has squared norm $L$ rather than $L/2$.

One standard sufficient form states that the Fourier series of a piecewise continuously differentiable periodic function converges at $x$ to $[f(x-)+f(x+)]/2$. At a continuity point it equals $f(x)$. This modern sufficient form should be distinguished from the exact hypotheses of the 1829 original.^[ref]

An even train of rectangular pulses of width $w<T$, period $T$ and unit area per pulse has Fourier coefficients $a_n=(2/T)\mathrm{sinc}(\pi nw/T)$ for $n\ge 0$. Here the sinc function is normalized as $\mathrm{sinc}z=\sin z/z$, and the constant contribution is $a_0/2=1/T$. Integrating the cosine over the pulse proves the formula; evenness eliminates the sine coefficients. At a pulse edge, pointwise convergence of a piecewise smooth Fourier series gives the mean of the one-sided values rather than a prescribed endpoint value.

For the Fourier series of a periodic rectangular pulse, the sinc function is nearly one when $nw/T\ll 1$ and has its first zero at $nw/T=1$. Thus narrowing a pulse increases the range of significant Fourier coefficients. Only integer harmonic indices occur, so an envelope zero need not be an exactly vanishing coefficient.

Extend $f(x)=\max (0,2x)$ on $(-1,1)$ with period two. Its Fourier coefficients in the basis $\cos (n\pi x),\sin (n\pi x)$ are $a_0=1$, $a_n=2[(-1{)}^n-1]/(n\pi {)}^2$, and $b_n=2(-1{)}^{n+1}/(n\pi )$, by integration by parts on $(0,1)$. ThusIts even part is the periodic extension of $|x|$ and its odd part is the Fourier series of a periodic sawtooth for $x$. The cosine terms converge absolutely and uniformly; the sine terms account for the jump at odd integers. At each jump, the full Fourier series has value $1$, the average of the two one-sided limits $0$ and $2$.

Extend $\cos (\mu x)$ from $[-\pi ,\pi ]$ with period $2\pi$. It is an even function, so all sine Fourier coefficients vanish. The product-to-sum identity givesIndeed, integrate $\cos (\mu x)\cos (nx)$ on $[0,\pi ]$ and use $\sin (\pi (\mu \pm n))=(-1{)}^n\sin (\pi \mu )$. ThusThe coefficients are $O(n^{-2})$, so the series converges absolutely and uniformly. The Dirichlet-Jordan convergence theorem identifies it with the periodic function everywhere, including the endpoints because the two endpoint values agree. Substituting a noninteger rational frequency and a convenient $x$ yields explicit series sums. Integer frequencies require taking removable limits rather than substituting into a zero denominator.

Let $\alpha ,\beta$ be nonzero real constants and let $f$ be a smooth periodic function of period $2\pi$, with real Fourier coefficients $c_n,d_n$. Write the unknown periodic solution with Fourier coefficients $a_n,b_n$. Fourier orthogonality applied to the equation givesSolving these two-by-two linear systems givesThe smooth forcing has rapidly decreasing Fourier coefficients, so the resulting series and its derivatives converge and satisfy the equation. Uniqueness follows because the difference of two solutions is $C{e}^{-\beta x/\alpha }$, which is periodic only when $C=0$. A single sinusoidal forcing therefore gives a single sinusoidal response with changed amplitude and phase.

Let $L>0$ and real $\kappa \ne 0$. Extend $e^{\kappa x}$ with period $2L$. In the Fourier series convention $a_0/2+{\sum }_{n\ge 1}[a_n\cos (n\pi x/L)+b_n\sin (n\pi x/L)]$, put $q_n=n\pi /L$. Direct integration givesFor example, the required antiderivatives are $e^{\kappa x}(\kappa \cos q_{n}x+q_n\sin q_{n}x)/({\kappa }^2+q_n^2)$ and $e^{\kappa x}(\kappa \sin q_{n}x-q_n\cos q_{n}x)/({\kappa }^2+q_n^2)$. The Dirichlet-Jordan convergence theorem gives the exponential value away from the jumps and their midpoint $\cosh (\kappa L)$ at odd multiples of $L$. For $\kappa =0$, the function is identically one, with $a_0=2$ and all other Fourier coefficients zero.

For $V(t)=V_0\max (\sin \omega t,0)$, with period $2\pi /\omega$, half-wave rectification givesThe Fourier coefficients are $a_0=2V_0/\pi$, $b_1=V_0/2$, $a_{2m}=-2V_0/[\pi (4m^2-1)]$, with all remaining coefficients zero. To derive the cosine coefficients, integrate $\sin s\cos ns$ over $0<s<\pi$ and use the product-to-sum identity. The sine coefficients follow from orthogonality. A half-period delay reverses only the fundamental sine term and leaves the constant and even-frequency cosine terms unchanged, by translation of a Fourier series.

For a period-$L$ Fourier series written as $a_0/2+{\sum }_{n\ge 1}(a_n\cos k_{n}x+b_n\sin k_{n}x)$, where $k_n=2\pi n/L$, translating the function by $\ell$ to $g(x)=f(x-\ell )$ rotates each pair of Fourier coefficients:The constant coefficient is unchanged. These formulas follow by the sine and cosine subtraction identities. In the complex Fourier series, the same statement is ${\hat{g}}_n=e^{-i{k}_n\ell }{\hat{f}}_n$. An integer-period translation changes no Fourier coefficient.

For $\beta >0$, periodically extend $\cosh (\beta x)$ and $\sinh (\beta x)$ from $-1<x\le 1$ with period $2$. Their Fourier series areand, away from odd integers,The first identity follows by integration of $\cosh (\beta x)\cos (n\pi x)$ on $[-1,1]$, using the antiderivativeFor the second identity, integration by parts and $d\cosh (\beta x)/dx=\beta \sinh (\beta x)$ show that its sine coefficient is $-n\pi /\beta$ times the first series' cosine coefficient. This avoids assuming that a Fourier series can always be differentiated term by term. The periodic hyperbolic cosine is continuous, so its series converges to it everywhere. The periodic hyperbolic sine has a jump discontinuity at each odd integer: the Dirichlet-Jordan convergence theorem gives the average of its two limiting values, namely zero, there. Both identities hold in the sense of mean-square convergence, which suffices for Parseval's identity.

A square wave alternates between two constant levels for equal halves of a period. Choose a $2\pi$-periodic function equal to $1$ on $(0,\pi )$ and $-1$ on $(-\pi ,0)$, with value zero at the jumps. Its odd function symmetry makes the mean and every cosine Fourier coefficient vanish. Directly integrating the two constant pieces givesThus only odd sine Fourier harmonics occur, with amplitudes decreasing like the reciprocal of their frequency multiplier. Every finite Fourier partial sum is smooth, yet their pointwise limit has jumps. At those jumps the Fourier series converges to zero, the midpoint of the two levels, regardless of which value we assign to the original function there.

This example explains how smooth components can encode sharp transitions. Increasing the number of Fourier harmonics narrows the transition region, but also produces the persistent overshoot described by the Gibbs phenomenon. The illustrated square wave and coefficient calculation are mathematical constructions, not measurements of a physical device.

A zero-mean square wave of amplitude $G$ has total mean-square signal power $G^2$. Each positive odd Fourier harmonic and its negative-frequency partner contribute $8G^2/({\pi }^2{r}^2)$, by its exact Fourier coefficients. The fundamental pair therefore accounts for $8/{\pi }^2$ of the total. The sum of contributions above harmonic index $R$ decays on the order of $R^{-1}$, although the discontinuous waveform still requires arbitrarily high harmonics for exact reconstruction.

The Gibbs phenomenon is the persistent overshoot near a jump discontinuity in ordinary truncated Fourier series. For this square wave, retain $M$ nonzero odd modes:As $M$ grows, the first peak approaches the jump while its height tends to approximately $1.17897974$. This is an excess of about $0.17898$ above the upper level $1$, or $8.949\%$ of the jump size $2$. The narrower region does not imply a vanishing maximum error. At the jump itself every $Q_M(0)=0$. These observations are compatible with pointwise convergence and $L^2$ convergence but exclude uniform convergence to the discontinuous square wave. ^[ref].

For this particular sum, its derivative is $Q_M^{\prime }(x)=(2/\pi )\sin (2Mx)/\sin x$ away from multiples of $\pi$, so the first positive peak is exactly at $x=\pi /(2M)$. There,This is a midpoint Riemann sum tending to $2\mathrm{Si}(\pi )/\pi$, where $\mathrm{Si}$ is the sine integral. It gives a short explanation of the nonzero limiting excess.

The phenomenon matters whenever a sharp boundary is represented by a limited set of Fourier harmonics: a visible ripple can be a property of the approximation rather than an oscillation in the original profile. Josiah Willard Gibbs reported the phenomenon in his 1899 letter “Fourier's Series” in Nature, volume 59, page 606. This is a documented historical report, without a claim of first discovery. ^[ref].

For real noninteger $\lambda$, the periodic extension from $[-\pi ,\pi ]$ has the displayed Fourier series, obtained by the product-to-sum identity in the sine coefficient integral. It equals the original function in the open interval and zero at the periodic endpoint jumps. At $\lambda =1/2$, Parseval's identity gives ${\sum }_{n\ge 1}{n}^2/(4n^2-1{)}^2={\pi }^2/64$. Integer frequencies instead already belong to the ordinary sine basis and should not be substituted into the removable coefficient singularities without taking limits.

The complex exponential functions $e^{int}$, for integers $n$, are orthogonal under the normalized integral inner product over a complete period. If $n=m$, the integrand $e^{i(n-m)t}$ is one; otherwise its antiderivative has equal endpoint values, giving zero. For the real modes, ${\int }_{-\pi }^{\pi }{\cos }^2(nt)dt={\int }_{-\pi }^{\pi }{\sin }^2(nt)dt=\pi$ for $n\ge 1$, while every mixed sine-cosine integral and every unequal-frequency pairing vanishes.

This makes the coefficient formulas useful: multiply a trigonometric polynomial by one mode and integrate, and all other modes cancel. For $f(x)=1/2+\cos x+\frac{1}{2}\sin (2x)$, integration against $\cos x$ gives $\pi$, and against $\sin (2x)$ gives $\pi /2$, recovering $a_1=1$ and $b_2=1/2$. Division by the squared norm of the mode supplies the normalization. A full period matters; truncating the integration interval generally destroys these cancellations.

If the symmetric partial sums of ${\sum }_{r\in \mathbb{Z}}{a}_r{e}^{irt}$ tend to zero at every point outside a finite set, then all coefficients vanish. Almost-everywhere convergence first gives $a_{\pm n}\to 0$. The twice-integrated series $F(t)=a_0{t}^2/2-{\sum }_{r\ne 0}{a}_r{e}^{irt}/r^2$ is continuous. Riemann summation of a convergent trigonometric series makes its symmetric second derivative zero away from the exceptional points, so it is affine on each intervening interval. Sinc-squared summation of a sequence tending to zero makes its symmetric first-order quotient zero everywhere, removing all possible affine corners. The globally affine function differs from a periodic function by $a_0{t}^2/2$, forcing $a_0=0$; periodicity then makes it constant, and integration against each nonzero frequency gives $a_r=0$.

Suppose the symmetric partial sums converge at a fixed point. Group positive and negative frequencies together, let $S_n$ be the resulting partial sums and set $w(x)=(\sin x/x{)}^2$, with $w(0)=1$. Summation by parts expresses the weighted sum as ${\sum }_{n\ge 0}{S}_n[w(nk)-w((n+1)k)]$, where $k=|h|/2$. The total variation of these weights is at most ${\int }_0^{\infty }|w^{\prime }(x)|dx<\infty$, independently of $k$, and each fixed difference tends to zero. Subtract the limiting value of $S_n$ and split into a finite prefix and a uniformly small tail to prove convergence to that value. This is a regular summation method even though the sinc-squared weights need not decrease monotonically.

If symmetric partial sums ${\sum }_{r=-n}^n{a}_r{e}^{irt}$ converge almost everywhere on the circle, then $a_n,a_{-n}\to 0$. Their successive differences $T_n=a_n{e}^{int}+a_{-n}{e}^{-int}$ tend to zero almost everywhere. If $M_n=|a_n|+|a_{-n}|$ failed to tend to zero, choose a subsequence with $M_n\ge \epsilon$. The normalized $T_n/M_n$ are bounded by one and tend to zero almost everywhere, so dominated convergence makes their squared integrals tend to zero. Fourier orthogonality instead makes those integrals $(|a_n{|}^2+|a_{-n}{|}^2)/M_n^2\ge 1/2$, a contradiction.

For $0<\epsilon \le 1/2$, this continuous function on $\mathbb{R}/\mathbb{Z}$ has nonnegative Fourier coefficientsIt is ${\epsilon }^{-1}$ times the periodic convolution of two indicators of $[-\epsilon /2,\epsilon /2]$. Its Fourier series is absolutely convergent, ${\sum }_r{\hat{b}}_{\epsilon }(r)=b_{\epsilon }(0)=1$, and ${\sum }_{|r|>R}{\hat{b}}_{\epsilon }(r)\ll 1/(\epsilon R)$. Consequently, if all $N$ points $t_n$ avoid the interval $\Vert t\Vert <\epsilon$, some frequency $1\le r\ll {\epsilon }^{-2}$ satisfies $|{\sum }_{n}e(r{t}_n)|\gg N\epsilon$. This detects missing small neighborhoods by a large exponential sum.

There are continuous functions whose Fourier series diverges at a prescribed point. In contrast, the Fejér kernel is nonnegative and has integral one, so Fejér summation is a uniform-norm contraction. The resulting Fejér sums need not be positive: for the constant function $f=-1$, every sum is $-1$.

A smooth compactly supported window with ${\sum }_{j\in \mathbb{Z}}\chi (x+j)=1$ permits multiplying every periodic distribution unambiguously. Its Fourier transform sampled at integer harmonics gives the periodic coefficients. The sharp window $H(x)-H(x-1)$ works for locally integrable functions but can be undefined at singular period endpoints, such as delta derivatives.

For the period-one continuation of $\log |x-1/2|$, symmetry and integration by parts give $c_m=-(-1{)}^m\mathrm{Si}(\pi m)/(\pi m)$ for $m\ne 0$. Thus the sine integral asymptotic gives $c_m=-(-1{)}^m/(2m)+1/({\pi }^2{m}^2)+O(m^{-4})$ as $m\to +\infty$. Translation of the logarithmic singularity causes the alternating phase; the endpoint derivative mismatch causes the next term.

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.

Every algebra character of the Wiener algebra is evaluation at a unique point of the unit circle. Writing $u(w)=w$, both $u$ and $u^{-1}$ have norm one. Thus $z=\varphi (u)$ satisfies $|z|\le 1$ and $|z^{-1}|\le 1$, hence $|z|=1$. Continuity then gives $\varphi ({\sum }_{n\in \mathbb{Z}}{a}_n{u}^n)=\sum a_n{z}^n$. Conversely, evaluation defines a continuous algebra character. The resulting bijection is a homeomorphism for the Gelfand topology.

The analytic Wiener algebra consists of the power series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ with ${\sum }_{n\ge 0}|a_n|<\infty$, identified with their values on the unit circle. The norm is $\Vert f\Vert =\sum |a_n|$. Discrete convolution gives multiplication and a submultiplicative norm; completeness follows from the absolutely summable sequence space. Each series converges uniformly on the closed unit disk and is holomorphic in its interior.

Every algebra character of the analytic Wiener algebra is evaluation at a unique point $z$ of the closed unit disk. Indeed, for $u(w)=w$, continuity and multiplicativity give $z=\varphi (u)$, $|z|\le \Vert u\Vert =1$ and $\varphi (f)={\sum }_{n\ge 0}{a}_n{z}^n$. Conversely, this formula defines a continuous algebra character for every such $z$. The evaluation correspondence is a homeomorphism for the Gelfand topology, since the power series are uniformly convergent on the closed unit disk.

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.

On a rectangle $(0,L_x)\times (0,L_y)$, products of the one-dimensional Fourier sine basis give an orthogonal basis of L2 space. Fourier orthogonality in each variable givesFor a constant function one, these coefficients are $4[1-(-1{)}^m][1-(-1{)}^n]/({\pi }^{2}mn)$: only odd indices occur. The product modes vanish on the rectangle's boundary, making them natural Dirichlet Laplacian eigenfunctions. In the heat equation, their amplitudes decay as $e^{-\kappa {\pi }^2(m^2/L_x^2+n^2/L_y^2)t}$. Initial data not vanishing at the boundary are still represented in the interior and in the square-integrable sense; the positive-time boundary values need not agree continuously with the initially assigned boundary values.

On $0\le x\le L$, with $L>0$, the tent $f(x)=\min (x,L-x)$ has Fourier sine series coefficients $b_n=4L\sin (n\pi /2)/({\pi }^2{n}^2)$. Split the coefficient integral at $L/2$ and integrate its two linear pieces by parts; their cosine endpoint terms cancel and their sine terms add. Reflection about the midpoint makes every even-index coefficient zero, and the remaining odd-index coefficients alternate in sign. The $n^{-2}$ bound gives absolute convergence and uniform convergence; the series equals the continuous tent, including its zero endpoints.

On $(0,1)$, integration by parts gives the displayed Fourier sine series. For normalized Dirichlet eigenfunctions $y_i=\sqrt{2}\sin [(i+1)\pi x]$, indexed from $i=0$, it has the form $A[y_0+{\sum }_{i\ge 1}{c}_i{y}_i]$, with $A=4\sqrt{2}/{\pi }^3$ and $c_i=[1-(-1{)}^{i+1}]/[2(i+1{)}^3]$. Its Rayleigh quotient for $-d^2/d{x}^2$ is $10$, since ${\int }_0^1(u^{\prime }{)}^{2}dx=1/3$ and ${\int }_0^1{u}^{2}dx=1/30$. Reflection symmetry forces $c_1=0$, because that first excited eigenfunction is antisymmetric about $1/2$. Thus the first excited state's contribution to the error is exactly zero; the first nonzero correction is $c_2=1/27$.