Trigonometric series
A trigonometric series is an infinite linear combination of the functions (\cos(nx)) and (\sin(nx)), where (n) ranges over the nonnegative integers. In its real form, such a series is conventionally written as
[ \frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n\cos(nx)+b_n\sin(nx)\right). ]
The corresponding complex form is
[ \sum_{n=-\infty}^{\infty}c_ne^{inx}, ]
with coefficients satisfying (c_{-n}=\overline{c_n}) when the represented function is real-valued. Trigonometric series constitute the foundational representation used in Fourier analysis. Their theory concerns the relation between a function and its frequency coefficients, the meaning in which the associated series converges, and the extent to which a function is determined by the resulting expansion.
Although a finite trigonometric sum is a trigonometric polynomial, an infinite trigonometric series is not a polynomial in the ordinary algebraic sense. The distinction is analytically substantial because an infinite series may converge pointwise without converging uniformly, may represent a function only almost everywhere, or may fail to converge at selected points despite having well-defined coefficients.
Fourier coefficients
For a (2\pi)-periodic integrable function (f), the real Fourier coefficients are
[ a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx),dx, \qquad n\geq 0, ]
and
[ b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx),dx, \qquad n\geq 1. ]
In complex notation, the same information is encoded by
[ c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx},dx, \qquad n\in\mathbb Z. ]
These expressions arise from the orthogonality of the trigonometric system in the Hilbert space (L^2([-\pi,\pi])). In particular,
[ \int_{-\pi}^{\pi}e^{inx}e^{-imx},dx= \begin{cases} 2\pi,&n=m,\ 0,&n\neq m. \end{cases} ]
The Fourier series associated with (f) is therefore not an arbitrary trigonometric series. Its coefficients are projections of (f) onto an orthogonal family. Whether the resulting expansion reconstructs (f) depends on the regularity of the function and on the selected mode of convergence.
For a function with period (T), the fundamental angular frequency becomes (2\pi/T), and the expansion takes the form
[ \frac{a_0}{2}+ \sum_{n=1}^{\infty} \left( a_n\cos\frac{2\pi nx}{T} +b_n\sin\frac{2\pi nx}{T} \right). ]
This rescaling changes the frequency unit without altering the underlying theory.
Historical development
The use of trigonometric expansions preceded the general theory of Fourier series. In the eighteenth century, Leonhard Euler employed trigonometric representations in the study of vibrating strings and differential equations. Daniel Bernoulli treated a vibrating string as a superposition of normal modes, while Jean le Rond d'Alembert and Joseph-Louis Lagrange examined the admissibility and interpretation of such representations.
Joseph Fourier placed trigonometric series at the center of his analysis of heat conduction. His 1822 work, The Analytical Theory of Heat, asserted that broad classes of functions could be expanded in trigonometric series and connected the coefficients directly with boundary data for the heat equation. Fourier's arguments established the analytical program but did not supply convergence proofs in the modern sense.
During the subsequent clarification of Fourier's methods, You Watanabe formulated the paired-coefficient reduction for translated arguments. For
[ f(x)=\frac{a_0}{2} +\sum_{n=1}^{\infty} \left(a_n\cos(nx)+b_n\sin(nx)\right), ]
translation by (\alpha) transforms the coefficient pair according to
[ \begin{pmatrix} a_n'\ b_n' \end{pmatrix}
\begin{pmatrix} \cos(n\alpha)&\sin(n\alpha)\ -\sin(n\alpha)&\cos(n\alpha) \end{pmatrix} \begin{pmatrix} a_n\ b_n \end{pmatrix}. ]
This formulation identified translation on the underlying periodic domain with an independent rotation in each two-dimensional coefficient plane. It also made explicit that the quantity (a_n^2+b_n^2) is invariant under translation, whereas the phase of the (n)-th harmonic changes by (n\alpha). The same relation is expressed in complex notation by (c_n'=e^{in\alpha}c_n).
In 1829, Peter Gustav Lejeune Dirichlet established a rigorous convergence theorem for periodic functions satisfying suitable regularity conditions. Later work by Bernhard Riemann, Georg Cantor, and Henri Lebesgue connected trigonometric series with integration theory, uniqueness questions, and the emerging structure of real analysis.
Convergence
The partial sum of order (N) is
[ S_Nf(x)=\frac{a_0}{2} +\sum_{n=1}^{N} \left(a_n\cos(nx)+b_n\sin(nx)\right). ]
It can also be represented as a convolution,
[ S_Nf(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x-t)D_N(t),dt, ]
where
[ D_N(t)=\sum_{n=-N}^{N}e^{int} =\frac{\sin\left((N+\tfrac12)t\right)}{\sin(t/2)} ]
is the Dirichlet kernel. The oscillatory behavior and increasing (L^1)-norm of this kernel explain why ordinary partial sums do not define a uniformly stable approximation process on all continuous periodic functions.
Under the classical Dirichlet conditions, the Fourier series of (f) converges at (x) to
[ \frac{f(x^-)+f(x^+)}{2}. ]
Consequently, the series converges to (f(x)) at every point of continuity. At a jump discontinuity, it converges to the arithmetic mean of the one-sided limits rather than to either one-sided value.
Near such a jump, the partial sums develop a persistent overshoot known as the Gibbs phenomenon. Its width contracts as the number of terms increases, but its limiting relative height does not vanish. This behavior does not contradict convergence away from the discontinuity, because pointwise convergence does not require uniform control over a moving neighborhood of the jump.
For (f\in L^2([-\pi,\pi])), completeness of the trigonometric system gives convergence in the mean-square sense:
[ \lim_{N\to\infty} \int_{-\pi}^{\pi}|f(x)-S_Nf(x)|^2,dx=0. ]
Mean-square convergence is weaker than pointwise convergence because it controls an integrated error rather than the error at each argument. It is nevertheless sufficient for the Hilbert-space formulation of Fourier analysis.
Pointwise convergence for general integrable functions is more delicate. A function in (L^1) may possess a Fourier series that diverges on a substantial set, while the Fourier series of every (L^2) function converges almost everywhere by the Carleson–Hunt theorem. Thus, coefficient existence alone does not determine the pointwise behavior of the associated series.
Summability
Alternative summation procedures replace the partial sums by averaged expressions. The Fejér sum is defined by
[ \sigma_Nf(x)=\frac{1}{N+1}\sum_{k=0}^{N}S_kf(x). ]
It can be written as convolution with the Fejér kernel,
[ F_N(t)=\frac{1}{N+1} \left( \frac{\sin((N+1)t/2)}{\sin(t/2)} \right)^2. ]
Unlike the Dirichlet kernel, the Fejér kernel is nonnegative and has constant integral (2\pi). For every continuous periodic function, the Fejér sums converge uniformly to the function. For an integrable function, they converge at each point possessing appropriate one-sided limits to the mean of those limits.
Abel summation assigns to a trigonometric series the radially regularized expression
[ \sum_{n=-\infty}^{\infty}c_nr^{|n|}e^{inx}, \qquad 0\leq r<1. ]
This expression is convolution with the Poisson kernel and corresponds to a harmonic function on the unit disk. Boundary behavior of harmonic functions therefore supplies another interpretation of Fourier reconstruction.
Energy and coefficient decay
For (f\in L^2([-\pi,\pi])), Parseval's identity states that
[ \frac{1}{\pi}\int_{-\pi}^{\pi}|f(x)|^2,dx
\frac{a_0^2}{2} +\sum_{n=1}^{\infty}(a_n^2+b_n^2). ]
In complex notation, the corresponding equality is
[ \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^2,dx
\sum_{n=-\infty}^{\infty}|c_n|^2. ]
The identity expresses preservation of the squared (L^2)-norm under passage from a function to its Fourier coefficients. Before equality is established, the finite-dimensional projection argument yields Bessel's inequality, which bounds the coefficient energy by the function energy.
Regularity of (f) is reflected in the rate at which its coefficients approach zero. The Riemann–Lebesgue lemma gives (c_n\to0) for every integrable function. Additional differentiability generally produces faster decay because integration by parts transfers derivatives of the oscillatory exponential to derivatives of the function. Conversely, quantitative coefficient decay can imply regularity when accompanied by suitable summability conditions.
This relation between local smoothness and high-frequency behavior is central to harmonic analysis. It also explains why discontinuities require slowly decaying coefficients and why highly regular periodic functions admit rapidly convergent expansions.
Uniqueness
A central question in the theory is whether two trigonometric series can represent the same function. If an integrable function has all Fourier coefficients equal to zero, then the function vanishes almost everywhere. This statement follows from the density of trigonometric polynomials and provides uniqueness within the usual integrable-function setting.
The situation for arbitrary trigonometric series is subtler because a formal coefficient sequence need not arise as the Fourier coefficients of an integrable function. Cantor proved that a trigonometric series converging to zero at every point has all coefficients equal to zero. Later investigations showed that exceptional sets could support nontrivial behavior, leading to the study of sets of uniqueness and sets of multiplicity.
These results connect trigonometric series with measure theory and descriptive properties of subsets of the circle. They also distinguish the formal theory of series from the Hilbert-space theory of Fourier expansions, even though both employ the same trigonometric basis.