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A trigonometric polynomial of degree at most $n$ obeys the displayed derivative bound. Iteration gives $\Vert T_n^{(r)}{\Vert }_{\infty }\le n^r\Vert T_n{\Vert }_{\infty }$. Applied to differences of approximants at dyadic degrees, this inequality underlies the inverse theorem for trigonometric approximation. It differs from the endpoint-weighted Bernstein inequality for algebraic polynomials.

A positive-definite function $r$ on the discrete group $\mathbb{Z}$ is the sequence of Fourier coefficients of a unique finite positive Borel measure on the circle, with total mass $r(0)$. Positivity makes the finite sums ${\sum }_{|n|<N}(1-|n|/N)r(n)e^{-2\pi int}$ nonnegative. Their measures converge weakly to the required measure; uniqueness follows from density of trigonometric polynomials in continuous functions.

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

A trigonometric polynomial is real on the unit circle precisely when its finite coefficients have the displayed symmetry. To prove necessity, conjugate its values on the circle, use $\overline{z}=z^{-1}$, and compare coefficients. Multiplying a coefficient difference by $z^d$ gives an ordinary polynomial vanishing at infinitely many points, so it is zero. Symmetry also gives $p(z^{-1})=\overline{p(\overline{z})}$ for every nonzero complex $z$.

Cosine substitution is an isometry from continuous functions on $[-1,1]$ to even continuous $2\pi$-periodic functions. It satisfies $\omega (\tilde{f},\delta )\le \omega (f,\delta )$ because cosine is Lipschitz continuous with constant one. Even degree-at-most-$n$ trigonometric polynomials correspond exactly to algebraic polynomials via $T_j(\cos \theta )=\cos (j\theta )$. Averaging a periodic approximant with its reflection cannot increase its error against an even target, so the algebraic and periodic best errors are equal.

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.

For the unit ball of $C^p({\mathbb{T}}^n)$ measured in the uniform metric, $c{\epsilon }^{-n/p}\le H(\epsilon )\le C{\epsilon }^{-n/p}\log (1/\epsilon )$. The lower bound uses disjoint bump functions of spatial scale $h$, height proportional to $h^p$ and independently chosen zero/one coefficients: there are $\exp (c{h}^{-n})$ functions separated at scale $h^p$. For the upper bound, Multivariable Jackson approximation gives a trigonometric polynomial of degree $O({\epsilon }^{-1/p})$. Its $O({\epsilon }^{-n/p})$ coefficients are bounded and can be rounded with coefficient accuracy proportional to $\epsilon$ divided by the coefficient count. Counting those rounded coefficient arrays supplies the logarithmic factor. The continuous-function unit ball without derivative bounds has infinite entropy for $\epsilon <1/2$, by an infinite sequence of disjoint height-one bumps.

The sequence $r(u,v)$ is equidistributed in ${\mathbb{T}}^2$ precisely when $ku+\ell v\notin 2\pi \mathbb{Z}$ for every nonzero integer pair $(k,\ell )$. Equivalently $1,u/(2\pi ),v/(2\pi )$ are rationally independent. Under that condition, nonconstant character averages tend to zero by the geometric-series formula. Uniform approximation by trigonometric polynomials gives convergence for continuous functions, and continuous upper and lower bounds give convergence for rectangles. If a character is resonant, the orbit lies in its proper closed kernel and misses a positive-area rectangle, so equidistribution fails.

Use the convention $K_m={\sum }_{|j|\le m}(1-|j|/(m+1))e^{ijt}$. Multiplying $K_{n-1}$ by $2n\sin (nt)$ gives Fourier coefficients $-ik$ for $|k|\le n$. A trigonometric polynomial of degree at most $n$ has no other frequencies, so convolution proves the displayed identity. Positivity and integral one of the Fejér kernel then give $\Vert P^{\prime }{\Vert }_{\infty }\le 2n\Vert P{\Vert }_{\infty }$. This is a useful elementary estimate; the sharp Bernstein inequality for trigonometric polynomials improves $2n$ to $n$.

A trigonometric polynomial nonnegative on the unit circle has a polynomial modulus-square factor of polynomial degree at most its trigonometric order. For a nonzero trigonometric polynomial of actual order $d$, conjugate symmetry of trigonometric polynomial coefficients yields reciprocal-conjugate root pairing for $P=z^{d}p$. Even multiplicity of unit-circle roots of a nonnegative trigonometric polynomial permits pairing all $2d$ roots of a polynomial. Select representatives ${\zeta }_1,\dots ,{\zeta }_d$ and useIt follows that $p(z)=c{\prod }_i|z-{\zeta }_i{|}^2$ with $c=(-1{)}^d{p}_d/{\prod }_i\overline{{\zeta }_i}$. Evaluating away from the roots of a polynomial gives $c>0$. Thus $q(z)=\sqrt{c}{\prod }_i(z-{\zeta }_i)$ works. Constant and zero trigonometric polynomials have constant or zero factors. Different selections of roots of a polynomial and constant phases can give different factors.

For the trigonometric polynomial $F(t)={\sum }_{k=1}^N{b}_{k}e(kt)$, orthogonality of complex exponentials givesIndeed, expand each square and use ${\int }_0^{1}e(jt)dt=0$ for every nonzero integer $j$, and one for $j=0$.

Every continuous $2\pi$-periodic function admits degree-at-most-$n$ trigonometric polynomials with supremum norm error bounded by a universal constant times its modulus of continuity at scale $1/n$, for $n\ge 1$. This first-order Jackson-type estimate transfers to algebraic polynomials through cosine substitution for polynomial approximation.

Fourier analysis describes a function by the oscillations that compose it. A graph records values at successive positions or times; a frequency spectrum records how much of each frequency is present, together with its wave phase. The same complicated shape can become a short list of simple components. For periodic functions, the components form a Fourier series; on the whole real line, the Fourier transform uses a continuous range of frequencies, and the Fourier inversion theorem puts them back together.

For a concrete example, consider the trigonometric polynomialWith $t$ measured in seconds, its three Fourier harmonics have frequencies $1$, $3$, and $5$ cycles per second and amplitudes $1$, $1/2$, and $1/4$. Each cosine has zero wave phase. Adding them creates the sharper peaks in the figure. The frequency spectrum makes the recipe immediately visible; changing an amplitude changes the shape without changing the component's frequency. Fourier orthogonality lets us recover each Fourier coefficient by averaging against the corresponding sine or cosine, because the other Fourier harmonics cancel over a full period of a function.

The deeper reason these components are so useful is that they simplify operations. By Euler's formula, the exponential function $e^{i\omega x}$ packages a cosine and a sine into one Fourier mode. A translation or a derivative changes only its multiplier:Thus Fourier modes are simultaneous eigenfunctions of these operations. In Fourier series, this gives a literal diagonalization in the Fourier basis; on the real line, the Fourier transform gives the continuous counterpart, with each frequency handled separately. The Translation property of the Fourier transform and the Fourier transform of a derivative turn shifting and differentiating a whole function into multiplication of its Fourier transform. A convolution likewise becomes multiplication by the convolution theorem. This is beautiful because one description simplifies several apparently different operations at once.

For example, the heat equation $u_t=\kappa u_{xx}$, with thermal diffusivity $\kappa >0$, evolves a spatial Fourier mode asFine ripples have larger angular frequencies and decay faster: the formula explains why heating smooths a jagged temperature profile. For a general initial profile with sufficient regularity, the Fourier transform gives $\hat{u}(\omega ,t)=e^{-\kappa {\omega }^{2}t}\hat{u}(\omega ,0)$, and Fourier inversion recovers the temperature. Similarly, multiplying selected parts of a frequency spectrum suppresses unwanted components. Claude E. Shannon's Communication in the Presence of Noise (1949), Section III, describes ideal filtering as a projection that retains some frequency coordinates and removes others.^[ref]

The physical motivation is already visible in Joseph Fourier's Théorie analytique de la chaleur (1822). Chapter III constructs heat profiles using sums of sines and cosines; Chapter IX, articles 355–359, passes from Fourier series to integrals as the spatial interval becomes unbounded. These heat problems explain the motivation for the discrete and continuous descriptions. Their modern formulations specify which functions admit reconstruction and in what sense the expansions converge.^[ref][ref]

For a finite complex measure on the additive circle group $\mathbb{R}/(2\pi \mathbb{Z})$, its Fourier coefficients are integrals against the group characters $e^{-inx}$, $n\in \mathbb{Z}$. They obey $|\hat{\mu }(n)|\le \Vert \mu \Vert$, where the norm is the total variation norm of a measure. A probability measure has zeroth coefficient one. The normalized Haar measure $m(dx)=dx/(2\pi )$ has all nonzero-frequency coefficients zero. A bounded family of finite Radon measures converges in the weak-star topology exactly when its coefficients converge to those of the proposed limit: extend linear combinations to all continuous tests using the density of trigonometric polynomials and the common variation bound.

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.

A Fourier partial sum retains the modes whose integer frequency multipliers satisfy $|k|\le n$ in a complex Fourier series:For a real function of period $2\pi$, this is equivalently $a_0/2+{\sum }_{k=1}^n[a_k\cos (kx)+b_k\sin (kx)]$. It is a trigonometric polynomial, even when the original function is discontinuous. For a square wave, retaining $M$ nonzero odd sine modes gives $S_{2M-1}f$; all even-frequency coefficients are zero. The Gibbs phenomenon illustrates why more retained modes need not reduce the maximum error.

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].

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.

The normalized even kernelis nonnegative, has integral one, and is a trigonometric polynomial of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\le C\min (n,n^{-3}|t{|}^{-4})$ for $|t|\le \pi$. This controls a weighted second moment and produces the Jackson operator estimate.

Define $h(x)=|x|\log (e\pi /|x|)$ on $[-\pi ,\pi ]$, set $h(0)=0$, and extend periodically. Concavity and monotonicity on $[0,\pi ]$ give $\omega (h,\delta )=\delta \log (e\pi /\delta )$ for $0<\delta \le \pi$. Its cosine Fourier coefficients areOne integration by parts gives the formula; the remaining integral is $\log j+O(1)$ by the Dirichlet test. All these coefficients are nonpositive. The de la Vallée Poussin sum $V_n=2{\sigma }_{2n}-{\sigma }_n$ reproduces degree-at-most-$n$ trigonometric polynomials and has operator norm at most three. Hence $L_n(f)=f(0)-V_{n}f(0)$ has norm at most four and annihilates those polynomials. Its weights on coefficients are zero up to $n$, nonnegative thereafter, and one from $2n$ onwards, so $|L_n(h)|\ge {\sum }_{j\ge 2n}|a_j|\gtrsim \log n/n$. Thus $E_n(h)\ge |L_n(h)|/4\gtrsim \log n/n$; the first Jackson theorem for periodic approximation proves the matching upper bound. This cusp and the critical Weierstrass modulus have the same first-modulus order but different best-approximation rates.