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When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples of the continuous representative of $f$, with the sign of the index reversed. Projecting $F$ in $L^2[-\pi ,\pi ]$ and then applying the Fourier inversion theorem gives the displayed reconstruction using the normalized sinc function, with each quotient defined as one when $t=n$. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the vector of shifted sinc functions, giving absolute uniform tails when the sample sequence belongs to ${\ell }^2$.

For $u_{xx}+y^2{u}_{yy}=0$ with zero vertical-side values on $0<x<\pi$, the Fourier coefficient $b_n(y)$ obeys the Cauchy-Euler differential equation $y^2{b}_n^{\prime \prime }=n^2{b}_n$. Boundedness at $y=0$ removes the negative indicial root, leaving $b_n(y)=A_n{y}^{(1+\sqrt{1+4n^2})/2}$. Bounded top coefficients give absolute convergence of the Fourier sine series below the top boundary.

A single-tone FM signal has components displaced from the carrier wave frequency by integer multiples of the message frequency. These are temporal sidebands; the spatial sidebands used to perturb a spatial pattern are a related but different setting. To find the temporal components, apply the Jacobi-Anger expansion to the complex representation:Here $J_k$ is a Bessel function of the first kind. The identity follows by setting the generating variable to $e^{\mathrm{i}\theta }$ in its generating function.^[ref]

Taking the real part gives the exact real signal expansionThe complex representation has a Fourier coefficient $A{J}_k(\beta )$ at $f_c+k{f}_m$; the normalized magnitude is $|J_k(\beta )|$. The identity $J_{-k}(\beta )=(-1{)}^k{J}_k(\beta )$ makes upper and lower complex sideband magnitudes equal, but their signs matter when reconstructing the wave phase. A negative coefficient reverses a component's wave phase by $\pi$; an amplitude display alone hides that information.

For $\beta =5$, original evaluation gives $J_0(5)\approx -0.17760$, $J_1(5)\approx -0.32758$, $J_5(5)\approx 0.26114$ and $J_6(5)\approx 0.13105$. Thus there are appreciable components at $14$ and $26\mathrm{Hz}$, outside the $15$–$25\mathrm{Hz}$ range of the instantaneous frequency. There is no contradiction: instantaneous frequency describes the local rate of the combined wave phase, while the frequency spectrum describes a global superposition of constant-frequency components.

For the real signal, negative frequencies must also be accounted for: a cosine at $-f$ equals one at $f$. When $f_c/f_m=20$, the normalized positive-frequency cosine coefficient at $n{f}_m$, $n>0$, is exactly $J_{n-20}(\beta )+J_{-n-20}(\beta )$. The second term is extremely small over the plotted range, but the figure includes it. The chosen complex representation need not have a frequency spectrum supported only at positive frequencies, even when its specified instantaneous frequency is everywhere positive.

For a nonzero FM modulation index, the ideal frequency spectrum has infinitely many sidebands. Many become tiny, but drawing only visible lines does not make the exact signal bandwidth finite. A useful engineering width must state a threshold or a retained-power fraction. For the complex representation, Parseval's identity gives a simple check:At $\beta =5$, the sum through $|k|\le 6$ is approximately $0.99356$. Retaining those complex components captures over $99\%$ of their squared magnitudes and spans $12\mathrm{Hz}$ from $14$ to $26\mathrm{Hz}$. This is a stated truncation criterion, not an exact cutoff or a universal bandwidth rule. It also explains why a diminished carrier component need not imply a diminished envelope: its contribution has been redistributed among sidebands.

A denominator such as $k\cdot \omega$ which may become small for large integer vectors $k$. Inverting a directional derivative on a torus divides the nonzero Fourier coefficients by $ik\cdot \omega$. A Diophantine condition bounds this loss polynomially, so exponential Fourier decay still gives convergence on a smaller complex strip.

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.

For an integrable periodic function of period $P>0$, its integral over an interval of length $P$ is independent of the starting point. Write $a=mP+r$ with $m$ an integer and $0\le r<P$. By a change of variables and the defining identity of a periodic function,No differentiability is required. This also allows the Fourier coefficients of a periodic function to be computed on any interval of one full period.

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.

Put $C={\hslash }^2/(2m)$, $v=|V_G|$, $\mathbf{G}=(2K,0)$, and $k_x=K+q$. The upper eigenvalue of the two coupled plane waves isAt fixed Fermi energy $C{k}_F^2$, its closed Fermi contour hasThus $A(v)=4{\int }_0^{q_m}y(q)dq$ can be inverted numerically to estimate the Bragg scattering coupling. The pocket disappears at $v=C(k_F^2-K^2)$. The estimate assumes a fixed chemical potential and neglects coupling to other plane waves; a measured pocket area alone does not determine all Fourier coefficients of an arbitrary lattice potential.

A continuous self-map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. An irrational rotation of the circle is uniquely ergodic: invariance multiplies its nonzero-index Fourier coefficients by nontrivial phases, forcing those coefficients to vanish. The Stone-Weierstrass theorem then identifies the invariant measure as normalized Lebesgue measure.

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.

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.

For a finite positive Borel measure on the circle, averaged squared Fourier coefficients tend to the sum of squares of its atomic masses. Express each squared coefficient as a double integral and average the circle character. The averaged character is bounded by one, tends to one on the diagonal and to zero elsewhere, so the dominated convergence theorem gives the diagonal mass. In particular an atomless measure has vanishing averaged absolute coefficients by the Cauchy-Schwarz inequality.