Fourier-Haar series

Haar–Fourier series

Consider an interval $(a,b)$, a measure $\mu$ on it and a corresponding complete orthonormal system of functions $\phi_0,\phi_1$ (so that $\int_a^b\phi_k(x)\phi_l(x)\,d\mu(x)=\delta_{kl}$). The Fourier series of a function $f$ with respect to such an orthonormal system of functions is:

\begin{equation}\sum_{k=0}^\infty c_k\phi_k,\label{a1}\end{equation}

with coefficients

$$c_k=\int\limits_a^bf(x)\phi_k(x)\,d\mu(x).$$

See Fourier series; Fourier series in orthogonal polynomials.

Depending on the orthonormal system used, one thus obtains

Fourier–Bessel series (see also Bessel functions);

Fourier–Chebyshev series (see also Chebyshev polynomials);

Fourier–Franklin series (see also Franklin system);

Fourier–Haar series (see also Haar system);

Fourier–Jacobi series (see also Jacobi polynomials);

Fourier–Laguerre series (see also Laguerre polynomials);

Fourier–Legendre series (see also Legendre polynomials);

Fourier–Walsh series (see also Walsh system). There are corresponding notions of coefficients, expansions and transforms (i.e., Fourier–Bessel coefficients, Fourier–Chebyshev coefficients, Fourier–Franklin coefficients, Fourier–Haar coefficients, Fourier–Jacobi coefficients, Fourier–Laguerre coefficients, Fourier–Legendre coefficients, Fourier–Walsh coefficients, etc.).

The properties of these coefficients and the convergence properties of the series \eqref{a1} often differ sharply from those in the trigonometric case; see, e.g., Fourier–Bessel integral; Laguerre transform.

There is a fair amount of variation in the terminology used: names can be switched and sometimes the word "Fourier" is left out altogether.