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The period average extracts the reference level of an integrable function that repeats. For a real- or complex-valued periodic function $f$ with positive period of a function $T$, assume $f$ is Lebesgue integrable on every bounded interval. Its average isThe answer is independent of the starting point $a$. Splitting a shifted interval at the old endpoint and using $f(t+T)=f(t)$ moves its extra piece back into the missing piece. The interval length must be a period; an arbitrary window can give a different average.

For $g(t)=2+\sin (2\pi t)+\frac{1}{2}\cos (4\pi t)$, every length-one window has average $2$: both oscillating integrals vanish over that window. A half-unit window instead has average $2+(2/\pi )\cos (2\pi a)$, which depends on its start. Thus a short sample of an oscillation can bias an estimate of its reference level even when the full-cycle average is known exactly.

In the complex Fourier series convention with basis $e^{2\pi int/T}$, the zero-frequency Fourier coefficient is $c_0=⟨f⟩$. This assertion defines the coefficient even when an additional convergence theorem is needed to reconstruct $f$ from its series. Strang's §4.1 explicitly identifies that constant coefficient with the average.^[ref]

Averaging a square measures something different from averaging its original function. For the illustrated $g$, orthogonality of the trigonometric terms gives $⟨g^2⟩=4+1/2+1/8=37/8$, whereas $⟨g{⟩}^2=4$. Its root mean square is therefore $\sqrt{37/8}$. Similarly, the period average of ${\sin }^2$ is $1/2$. A zero mean need not imply a zero mean square.

A periodic convolution operator with continuous periodic kernel is compact on $L^2[0,2\pi ]$, since its kernel is square integrable. In the normalized Fourier series basis it is diagonal, with multipliers $c_n={\int }_0^{2\pi }K(x)e^{-inx}dx$. These multipliers include the full integral rather than its normalized Fourier coefficient.

For a circle of circumference $2\pi$, the heat equation $u_t=D{u}_{xx}$ has periodic matching of both $u$ and $u_x$ at $x=\pm \pi$. For $D>0$, the displayed Fourier series solves the initial-value problem, with $\bar{f}=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }f(x)dx$ and the usual real Fourier coefficients. Every nonconstant normal mode decays, leaving the conserved mean. In particular $f(x)=(\pi -|x|{)}^2$ gives $\bar{f}={\pi }^2/3$, $a_n=4/n^2$ and $b_n=0$. The initial cusp is smoothed for every positive time.

A distribution has period $T>0$ if the displayed equality holds for every test function $\varphi$. This agrees with $f(x+T)=f(x)$ almost everywhere for a regular distribution. If $u$ is a tempered distribution, density of test functions in the Schwartz space extends the identity to all Schwartz functions.

For a test function $\Psi$ with ${\sum }_k\Psi (x+kT)=1$, one hasIndeed, expand $\varphi ={\sum }_k\Psi (x+kT)\varphi (x)$ in the Schwartz space and move each translation to $\varphi$ by periodicity. All Schwartz seminorms of the localized terms decay faster than any power of $|k|$, justifying the sum of pairings. This localization permits rigorous Fourier coefficients without multiplying a distribution by a discontinuous window.

For $T>0$, values of a function $f$ on a half-open interval $[a,a+T)$ determine its periodic extension to the real line: choose the unique integer $k$ with $x-kT\in [a,a+T)$ and set $F(x)=f(x-kT)$. The resulting periodic function has period $T$. If the endpoint limits of $f$ differ, the extension has a jump discontinuity at every period boundary. Under the usual piecewise-smooth convergence hypotheses, its Fourier series takes the average of the two one-sided limits there; an assigned value at an isolated endpoint does not change the Fourier coefficients.

A periodic function repeats exactly after a nonzero shift of its input. For $f:\mathbb{R}\to Y$, a positive period of a function $T$ satisfiesOne interval of length $T$ therefore determines the whole function. For another domain of a function $D$, a period must also preserve the domain, $D+T=D$; equality only where two domains happen to overlap is weaker. Periodicity is an exact translation of a function symmetry, rather than merely two similar-looking pieces of its graph.

The sine $\sin (2\pi x)$ has fundamental period one. The function ${\sin }^2(2\pi x)$ repeats after half a unit: the sine changes sign under that shift, but squaring removes the sign. The mixed function $g(x)=2+\sin (2\pi x)+\frac{1}{2}\cos (4\pi x)$ again has fundamental period one. All positive integer multiples of a period are periods, so specifying one period need not identify the shortest repetition.

If the input is time and $T_0$ is the fundamental period, the frequency is $1/T_0$ and the angular frequency is $2\pi /T_0$. Repetition lets a Fourier series describe a cycle using sines and cosines; the period average separates the reference level from the oscillating terms. Strang develops this representation and its Fourier coefficients in §4.1 of Computational Science and Engineering by Gilbert Strang.^[ref]

This connection grew out of a physical problem. In his 1822 book Théorie analytique de la chaleur by Joseph Fourier, Joseph Fourier used periodic functions to extend profiles beyond their initial interval while treating heat conduction. Chapter III, §§233–234 combines even functions and odd functions into sine–cosine expansions and rescales their period.^[ref] A modern Fourier series interpretation still requires the appropriate convergence conditions.

A constant function has every real shift as a period and no fundamental period. Without continuity, even a nonconstant periodic function can lack a smallest positive period; periodic function without a least positive period gives a concrete example. The period group of a function organizes all these possibilities.

For a two-periodic function and $N$ equally spaced samples,ConsequentlyExponential Fourier coefficient decay therefore gives exponential convergence in $N$.

If a periodic unimodular function $a$ and its inverse have absolutely summable Fourier coefficients, replacing ${\hat{\phi }}_0$ by $\hat{\phi }=a{\hat{\phi }}_0$ preserves the orthonormal integer-translate span and integrability of an integrable scaling function. In physical space this is an absolutely summable combination of integer function translations. The new MRA low-pass filter is $m(t)=a(2t)m_0(t)/a(t)$. A rough phase can therefore preserve the multiresolution analysis while changing MRA low-pass filter regularity.

For $L>0$, extend $f_L(x)=\sin x-x\sin L/L$ from $[-L,L]$ periodically. The linear term makes both endpoint values zero and the endpoint first derivatives match. The function is odd, so its Fourier series has only sine terms, with the displayed Fourier coefficients. Indeed, for $k_n=n\pi /L\ne 1$,subtracting and multiplying by $2/L$ proves the formula. The coefficients decay as $O(n^{-3})$. At $L=m\pi$, the single coefficient $b_m$ has removable limit $1$, with all others zero. Multiplying $b_n$ by $e^{-\kappa k_n^{2}t}$ solves the corresponding heat equation on the circle.

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.

For a Schwartz function on the real numbers, its periodization $P_f$ is a smooth periodic function of period of a function one. Rapid decay gives uniform convergence on $[0,1]$ of the defining series and each differentiated series. Its $k$th Fourier coefficient isThe change of variable $u=x+n$ uses $e^{2\pi ikn}=1$. Rapidly decreasing Fourier coefficients give the absolutely and uniformly convergent Fourier series $P_f(x)={\sum }_k\hat{f}(k)e^{2\pi ikx}$. Evaluation at $x=0$ proves the Poisson summation formula.^[ref]

The same identity explains aliasing in the Nyquist–Shannon sampling theorem: applying the Poisson summation formula to the Schwartz function $x\mapsto f(x)e^{-2\pi i\xi x}$ givesThe sampled spectrum is periodic in frequency, and shifted copies of the continuous Fourier transform add together. When those copies overlap, the samples cannot separate their contributions. The Nyquist–Shannon sampling theorem adds a bandlimited function hypothesis to make reconstruction possible.

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.

For $0\le r<1$, this Poisson kernel for the unit disk has the uniformly convergent Fourier seriesThe identity follows by taking the real part of the geometric series $1+2{\sum }_{n\ge 1}(r{e}^{i\theta }{)}^n$. For integrable $2\pi$-periodic boundary data $f$, its Poisson extension iswhere $a_n,b_n$ are the usual Fourier coefficients. Each term is a harmonic function in the open disk. For continuous boundary data, the extension solves the Dirichlet problem with those boundary values.

For $0<r<1$, the geometric series gives $P_r(t)=1+2{\sum }_{n\ge 1}{r}^n\cos nt$. Hence this positive even kernel has normalized integral one, and normalized convolution with it multiplies the $n$th Fourier coefficient by $r^{|n|}$. Its mass outside any fixed neighbourhood of zero tends to zero as $r\uparrow 1$. If an integrable function has finite one-sided limits $A,B$ at zero, evenness gives half the mass to each side and the convolution tends to $(A+B)/2$.

For a linear phase $e^{2\pi i\theta x}$ on $[n]$, set $Q=\lfloor \sqrt{n}\rfloor$. The Dirichlet approximation theorem gives $1\le q\le Q$ with $\Vert q\theta {\Vert }_{\mathbb{R}/\mathbb{Z}}\le Q^{-1}$. If $1\le L$ and $2L\le Q$, split each residue chain modulo $q$ into arithmetic progressions of lengths between $L$ and $2L-1$, merging the final remainder into the preceding block. Each chain has at least $Q$ points, so this is possible. On each resulting arithmetic progression, the linear phase varies from its first value by at most $4\pi L/Q$. This converts a large Fourier coefficient of a zero-sum real function into positive mean on one block: freezing the linear phase controls the weighted block sums, and their positive and negative total masses agree.

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.

For $b>0$ and $-b\le y\le b$, the even quadratic polynomial $b^2-y^2$ has the displayed expansion in even Dirichlet eigenfunctions. Put $k_n=(2n+1)\pi /(2b)$. Their squared norm on $[-b,b]$ is $b$. Two integrations by parts, using $\cos (k_{n}b)=0$, giveDivision by $b$ gives the displayed Fourier coefficients. The cubic decay gives uniform convergence. The half-integer frequencies arise from requiring both evenness and zero values at the endpoints, rather than from a periodic boundary condition. This expansion is useful in rectangular duct flow.

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

A level-one modular form with rational Fourier coefficients is a weighted homogeneous polynomial with rational coefficients in the normalized Eisenstein series $E_4,E_6$. Subtract its constant coefficient times a same-weight monomial. The remainder is a cusp form; divide it by ${\Delta }_0=(E_4^3-E_6^2)/1728=q+O(q^2)$. The valence formula for the modular group shows that ${\Delta }_0$ has no interior zeros, so this quotient is holomorphic and has rational coefficients. Induction on weight proves the assertion. Euler's Bernoulli number formula for even zeta values then gives the corresponding rational weighted polynomial in the unnormalized lattice Eisenstein sums $G_4,G_6$.

A removable discontinuity has an existing finite limit that differs from the assigned point value, or a missing point value. Changing the point value removes it. Integrals and Fourier coefficients do not detect changes on a set of measure zero, so purely integral hypotheses cannot prohibit this kind of discontinuity for an arbitrary representative.