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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.

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.

Periodicity is invariance under translation by a nonzero period $P$: a periodic function satisfies $f(x+P)=f(x)$ at every point in its domain, with translation preserving the domain. For an angular coordinate on a full circle, single valuedness requires $2\pi$-periodicity. The sine and cosine factors $\sin (k\theta )$ and $\cos (k\theta )$ are $2\pi$-periodic for nonzero real $k$ precisely when $k$ is an integer. Their Fourier series supply the angular modes for the harmonic expansion on a disk.

A periodic solution is a solution of a differential equation which is a periodic function: for some $T>0$, $x(t+T)=x(t)$ at every time. This demands repetition of the state, rather than merely repetition of the coefficients in the differential equation. A nonconstant periodic solution of an autonomous differential equation traces a periodic orbit; constant equilibrium points are also periodic functions, so they must be excluded when asserting absence of periodic orbits.

For an explicit example, consider the initial value problemIts coefficient has period $T=2\pi$. Taking a logarithmic derivative and then applying integration givesDirect differentiation verifies both the differential equation and the initial condition. After one coefficient period,Thus the repeating coefficient generally produces a repeating shape with a changed amplitude. With $b=1$, the three cases $a=-0.2,0,+0.2$ have one-period factors approximately $0.2846,1,3.5136$, respectively. Only $a=0$ gives a periodic solution with this initial condition. For $a<0$ the positive solution of a differential equation tends to zero, and for $a>0$ it tends to infinity; neither can have any positive period. The zero solution of a differential equation, obtained with a different initial condition, is periodic for every $a$.

The exponential function separates two roles:For $b=1$, even the decaying case $a=-0.2$ initially rises, because $y^{\prime }(0)=0.8>0$. The growing case $a=0.2$ has intervals of decline, because $a+\cos t$ becomes negative. This is the distinction between the instantaneous logarithmic derivative $y^{\prime }/y$ and the Floquet growth rate measured over many periods.

More generally, for a real continuous periodic function $q$ of period $T$, the scalar linear differential equation $y^{\prime }=q(t)y$ hasA nonzero solution of a differential equation is $T$-periodic exactly when $m=1$, or equivalently when the coefficient has zero integral over a period. This scalar calculation is the simplest example of a Floquet multiplier.^[ref]

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.

Phase folding of a light curve maps timestamps onto a trial cycle of duration $P>0$. Given a reference time $t_0$, define the phase in cycles bywhere $\mathrm{frac}(x)=x-\lfloor x\rfloor$ is the fractional part, using the floor function also for negative $x$. Plot brightness against ${\varphi }_i$ rather than $t_i$; measurements separated by an integer multiple of $P$ occupy the same phase. For a periodic function $F(t+P)=F(t)$, this reveals the shared shape of many cycles. Folding changes coordinates; it neither averages samples nor chooses the correct $P$ by itself. Welsh et al.'s folded Kepler plot separates the exoplanet transit and orbital brightness modulation. ^[ref].

The second figure uses a completely synthetic trapezoidal dip: $P=3$ days, $t_0=1$ day, and the same $1\%$ depth and four-hour duration as the first figure. Different colours identify four successive cycles. Correct folding superimposes the dips; a wrong trial orbital period spreads them out. If $P^{\prime }=P+ϵ$, the centre of the $n$th repeated event shifts by approximately $-nϵ/P$ cycles when folded on $P^{\prime }$, for $|ϵ|\ll P$. With $P^{\prime }=3.03$ days, the fourth plotted event is displaced by $-3(0.03)/3.03\simeq -0.0297$ cycles. Transit-timing variations can also smear a constant-period fold, so retain the original timestamps and inspect individual events.

One may combine nearby phases with a weighted mean, but the uncertainty reduction requires assumptions. For $N$ independent measurements with common standard deviation $\sigma$ and effectively the same expected brightness within the bin, their arithmetic mean has standard deviation $\sigma /\sqrt{N}$. Thus sixteen independent $0.0006$-precision samples give $0.00015$ precision on that phase-bin mean. Shared calibration errors, changing starspots, and bins wide enough to mix ingress with out-of-transit brightness violate these assumptions. A folded plot can conceal changes between cycles. Gaussian process models can represent correlated variability and interpolate irregular samples, but their predictions and uncertainties depend on their assumed covariance function.

The bottom panels show the same model as relative brightness $f=F/F_0$ and as magnitude difference in one fixed photometric passband with a fixed magnitude zero point:A dip to $f=0.99$ gives $\Delta m\simeq +0.010912$ magnitudes: fainter means a larger apparent magnitude. The figure places larger magnitude differences lower on the page so the two dip shapes are easy to compare; reversing that plotting convention does not change the formula. For $f=1-\delta$ and small $\delta$, expansion of the logarithm gives $\Delta m\simeq (2.5/\ln 10)\delta \simeq 1.085736\delta$. The exact transform is nonlinear: transforming an exposure's mean brightness is generally different from averaging its instantaneous apparent magnitudes. The ordinary logarithm cannot be applied to a nonpositive background-subtracted measurement; keep such measurements and their uncertainties in brightness units for an appropriate statistical model.

A gravitational wave of plus polarization has opposite transverse strain or geodesic deviation eigenvalues along two fixed perpendicular axes. For a sinusoidal periodic function profile, the stretching and compressing roles alternate every half-period.

A quotient topological space is a quotient set equipped with the quotient topology. For a concrete gluing, give the closed interval $I=[0,1]$ its usual subspace topology, identify $0\sim 1$, and leave every interior point in its own singleton equivalence class. The resulting quotient space is a circle: the two endpoints become one point, while interior points remain distinct.

The exact continuous map to the unit circle in the plane isHere sine and cosine give $p(0)=p(1)=(1,0)$, and $p(s)=p(t)$ holds exactly when $s=t$ or $\{s,t\}=\{0,1\}$. Thus $p$ induces a bijection $\bar{p}:I/\sim \to S^1$. ^[ref] The universal property of the quotient topology makes $\bar{p}$ continuous. The quotient of a compact space is compact, and the unit circle is Hausdorff, so the compact-to-Hausdorff continuous bijection theorem makes $\bar{p}$ a homeomorphism. This verifies the topology, not merely the point identifications.

To inspect an open neighbourhood of the seam, choose $0<\epsilon <1/2$ and let $U_{\epsilon }=p([0,\epsilon )\cup (1-\epsilon ,1])$. Its full preimage isBoth pieces are open in the subspace topology on $I$, so $U_{\epsilon }$ is open in the circle. In contrast, $A=[0,\epsilon )$ is open in $I$, butis not open in $I$: it contains $1$ without any neighbourhood of $1$. Hence $p(A)$ is not open in the circle. A quotient map can therefore fail to be an open map.

For a continuous function $f:I\to \mathbb{R}$, the only gluing condition is $f(0)=f(1)$. For example, $f(t)=\cos (2\pi t)$ defines a continuous function on the circle, whereas $f(t)=t$ does not. This lets us use an interval as a coordinate for a periodic function while making the seam invisible. More general quotient spaces encode glued geometric pieces in the same way: describe continuous maps before gluing and check that their values agree on every equivalence class.

The Hausdorff property need not survive a quotient topology. In the non-Hausdorff quotient of the real line by rational translation, $x\sim y$ means $x-y\in \mathbb{Q}$. A nonempty open preimage $W$ must contain all translates by rational numbers of any open interval inside it. The density of the rational numbers then forces $W=\mathbb{R}$: for any $y$, some rational number $r$ has $y-r$ in that interval. Thus this quotient space has the indiscrete topology. The irrationality of the square root of two gives distinct equivalence classes for $0$ and $\sqrt{2}$, so this quotient space is not Hausdorff. Unlike the endpoint-gluing circle, this gluing identifies a dense subset with each point.

A point-sampling model assigns $p_{rc}=f(x_c,y_r)$ to a pixel on a grid, where $f$ is a continuous function representing an image signal. An area-sampling model instead assigns the average value of a function over a specified footprint. These are different models: a narrow feature between point samples can be missed, whereas its contribution to an area sample depends on its width and shade. Neither model says that the recorded value uniquely determines everything inside a grid cell.

The upper panels start with exactly the same $6\times 4$ array. Nearest-neighbor interpolation repeats its values into a $24\times 16$ display; successive linear interpolations in the two coordinate directions produce a smoother $24\times 16$ display. Original sample positions are marked. Both displays use 384 output pixels, but only the same 24 recorded values. The extra values follow the chosen interpolation rule. Between-grid appearance is inferred, not independently recorded.

Spatial resolution needs a stated meaning. A grid over a rectangle of width $L_x$ and height $L_y$ has nominal cell pitches $\Delta x=L_x/W$ and $\Delta y=L_y/H$. Over the same rectangle, doubling both dimensions halves these pitches and quadruples the pixel count. That is a finer grid, not a guarantee of more recoverable detail. Blurring before sampling, quantization and aliasing can make different originals share the same recorded array. Enlarging the rectangle without adding samples instead increases the displayed pitch.

A reproducible computational demonstration makes the ambiguity exact. Take the one-dimensional profiles $f(x)=0.5+0.4\sin (2\pi x)$ and $g(x)=0.5+0.4\sin (10\pi x)$, repeated identically along every row. Sample at $x_j=j/4$ for $j=0,\dots ,8$. Since $10\pi x_j=2\pi x_j+2\pi j$, the fact that the sine is a periodic function gives $f(x_j)=g(x_j)$. Both yield $0.5,0.9,0.5,0.1,0.5,0.9,0.5,0.1,0.5$, although at $x=1/8$ they give approximately $0.78284$ and $0.21716$. The lower panel draws both originals and their common samples: no operation on those samples alone can identify the original profile. This is aliasing in an ideal point-sampling model, without image noise or rounding.

The Nyquist–Shannon sampling theorem describes reconstruction under a bandwidth restriction and ideal sampling assumptions; it does not grant uniqueness for arbitrary image signals from a finite raster image.^[ref] The beautiful lesson is constructive: choose what detail the grid should preserve before sampling, and choose a reconstruction rule suited to the information actually retained.

For a real angle $\theta$, $\sin \theta$ is the vertical coordinate on the unit circle reached by a counterclockwise rotation through $\theta$ radians. For an acute angle in a right triangle, it is the opposite side divided by the hypotenuse. In particular,The first-quarter rise and second-quarter fall are visible in the circle projection; the lower semicircle gives negative values.

Reflecting the unit circle across the horizontal axis gives $\sin (-\theta )=-\sin \theta$, so sine is an odd function. A full turn gives $\sin (\theta +2\pi )=\sin \theta$, making it a periodic function. For real arguments, its range is $[-1,1]$ and its zeros of a function occur at $\theta =k\pi$ for integers $k$. The cosine describes the perpendicular coordinate of the same point, and the Pythagorean trigonometric identity keeps their two values tied together.

For each $T>0$ there is a nonnegative test function $\Psi$ with the displayed identity. Choose a nonnegative bump function $\eta$ supported in $[-T,T]$ and positive on $[-T/2,T/2]$. The sum $S(x)={\sum }_k\eta (x+kT)$ is a positive, smooth, $T$ periodic function. Setting $\Psi =\eta /S$ preserves compact support and smoothness, and periodicity of $S$ proves ${\sum }_k\Psi (x+kT)=S(x)/S(x)=1$. The translates form a locally finite partition of unity on the real line.

The Normal convergence of the Weierstrass elliptic-function series allows its derivative to be written as an absolutely locally uniformly convergent sum of inverse cubes. For a period lattice $\Lambda$, this derivative sum is invariant under translation by any lattice vector, by reindexing. Consequently $\wp (z+\omega )-\wp (z)$ is constant; its apparent poles cancel. The Weierstrass elliptic function is even, so for a primitive lattice generator $\omega$, evaluation at $z=-\omega /2$ makes that constant zero. Applying this to two generators proves that this is a periodic function, namely $\wp$. This argument avoids separating the original convergent series into two divergent inverse-square sums.

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.

A triangular wave is a continuous piecewise linear periodic function whose slope alternates between two values. The even $2\pi$-periodic extension of $bx$ from $[0,\pi ]$ is $P(s)=b|s-2k\pi |$ for $(2k-1)\pi \le s\le (2k+1)\pi$. Its Fourier cosine series contains a constant term and odd modes with coefficients proportional to the inverse square of the mode number. Such extensions describe reflection under Neumann boundary conditions.

Trigonometric functions turn an angle into a coordinate or a ratio of lengths. Start at $(1,0)$ on the unit circle and rotate counterclockwise through $\theta$ radians: the resulting point isThe cosine records horizontal position and the sine records vertical position. This definition continues around the whole circle, including negative angles, so the signs encode direction rather than requiring a separate rule for each right triangle.

The unit circle picture explains why these functions are useful beyond measuring triangles. The two coordinates of a point rotating at constant angular speed are periodic functions. Scaling the radius to $A$ and using $\theta =\omega t+\varphi$ gives $x=A\cos (\omega t+\varphi )$ and $y=A\sin (\omega t+\varphi )$. A circular motion becomes two simple repeating traces, connecting Euclidean geometry, simple harmonic motion, and Fourier series.

For an acute angle in a right triangle, rescaling its hypotenuse to one recovers the familiar ratios: sine is opposite side divided by hypotenuse, and cosine is adjacent side divided by hypotenuse. The tangent function is their ratio, $\tan \theta =\sin \theta /\cos \theta$, wherever the denominator is nonzero; the secant function, cosecant, and cotangent give the other reciprocal ratios.

The same names extend to complex numbers through the complex exponential:These definitions agree with the circle interpretation for real angles. For complex arguments, the values need not lie between $-1$ and $1$; the literal circle-coordinate picture is restricted to real arguments. ^[ref]