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The Ramanujan tau function gives the Fourier coefficients at positive indices of the modular discriminant: $\Delta (z)={\sum }_{n\ge 1}\tau (n)q^n$, with $\tau (1)=1$. The Serre derivative and vanishing of weight-fourteen level-one cusp forms give $qd\Delta /dq=E_2\Delta$ and hence the convolution recurrence

When the continuous Fourier transform $F$ of an integrable function vanishes outside $[-\pi ,\pi ]$, its periodic Fourier coefficients are the integer samples 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 sinc function reconstruction. The Cauchy-Schwarz inequality controls the reconstruction error uniformly in $t$. Bessel's inequality bounds the shifted sinc coefficient vector, 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 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 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.