View source for Fourier-Haar series

''Haar–Fourier series''

Consider an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301801.png" />, a [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301802.png" /> on it and a corresponding complete [[Orthonormal system|orthonormal system]] of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301803.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301804.png" />). The Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301805.png" /> with respect to such an orthonormal system of functions is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

with coefficients

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130180/f1301807.png" /></td> </tr></table>

See [[Fourier series|Fourier series]]; [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]].

Depending on the orthonormal system used, one thus obtains

[[Fourier–Bessel series|Fourier–Bessel series]] (see also [[Bessel functions|Bessel functions]]);

Fourier–Chebyshev series (see also [[Chebyshev polynomials|Chebyshev polynomials]]);

Fourier–Franklin series (see also [[Franklin system|Franklin system]]);

Fourier–Haar series (see also [[Haar system|Haar system]]);

Fourier–Jacobi series (see also [[Jacobi polynomials|Jacobi polynomials]]);

Fourier–Laguerre series (see also [[Laguerre polynomials|Laguerre polynomials]]);

Fourier–Legendre series (see also [[Legendre polynomials|Legendre polynomials]]);

Fourier–Walsh series (see also [[Walsh system|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 (a1) often differ sharply from those in the trigonometric case; see, e.g., [[Fourier–Bessel integral|Fourier–Bessel integral]]; [[Laguerre transform|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.