Trigonometric systems and Fourier coefficients

Let $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ and $d\mu=dx/(2\pi)$. For an integrable complex-valued function, define its coefficient at integer $n$ by pairing with $e^{inx}$. The negative sign in the coefficient integral is the complex conjugate required by the inner product. Translating a function changes coefficient phases, while multiplying by a character shifts frequency indices. A different period or an unnormalized measure changes formulas, so normalization must be recorded before applying Parseval or comparing numerical values.

Orthogonality follows by integrating $e^{i(n-m)x}$ over a full period: the result is one when the integers agree and zero otherwise. Real functions satisfy $c_{-n}=\overline{c_n}$, because conjugating the coefficient integral reverses the sign of the frequency. This symmetry does not require the coefficients themselves to be real. A sine contributes purely imaginary coefficients with opposite signs. Its complex coefficients therefore encode both amplitude and phase; replacing them by magnitudes alone loses information about the original signal.

For $f\in L^2(\mathbb T)$, the symmetric partial sum $S_Nf=\sum_{|n|\le N}c_ne^{inx}$ is the orthogonal projection onto those frequencies. Its error is perpendicular to every retained mode. Pythagoras shows that it minimizes squared error among trigonometric polynomials with that frequency range, and Bessel's inequality bounds the retained coefficient energy by the input energy. This is an exact Hilbert-space statement; it does not require the graph to look smooth or each pointwise error to shrink monotonically.

The exponentials are complete in this Hilbert space. One route to completeness first approximates continuous periodic functions by suitable trigonometric averages and then uses density of continuous functions in the square-integrable class. Completeness upgrades Bessel's inequality to Parseval and makes the symmetric projections converge in squared norm. The functions in this space are equivalence classes modulo null sets, so values changed at a single point do not alter any coefficient, energy, or norm convergence statement.

An integrable periodic function has coefficients tending to zero as frequency grows, by the Riemann–Lebesgue lemma. This necessary decay does not imply that their absolute sum is finite, and it does not by itself establish pointwise reconstruction. In the square-integrable setting the coefficient sequence is square summable, but square summability alone still leaves pointwise questions separate. Always name the target topology: convergence of energy error, convergence at a point, and uniform convergence make different claims about the same sequence of polynomials.

Finite sampling adds another distinction. A discrete Fourier transform of sampled values gives coefficients of a finite data vector, and unresolved frequencies can alias into lower ones. It is not automatically the same as the exact integral coefficients of a continuous signal. An analytic coefficient calculation provides a reference against which sampling and quadrature can be checked. When a model is band limited and the sampling conditions are supplied, the bridge can be made precise; without those conditions an attractive sampled spectrum can conceal frequencies that were never resolved.