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One of the classical orthonormal systems of functions. The Haar functions $\chi_n$  of this system are defined on the interval $[0,1]$ as follows:
$$
\chi_1(t) \equiv 1\quad \text{ on } [0,1];
$$

if $n=2^m+k$, $k=1,\dots, 2^m$, $m=0,1,\dots$, then
$$
\chi_n = \begin{cases} \sqrt{2^m} \quad &&\text{ for } t\in\left(\frac{2k-2}{2^{m+1}}, \frac{2k-1}{2^{m+1}}  \right),\\
-\sqrt{2^m} \quad &&\text{ for } t\in\left(\frac{2k-1}{2^{m+1}}, \frac{2k}{2^{m+1}}  \right),\\
0  \quad &&\text{ for } t\not\in\left(\frac{k-1}{2^{m}}, \frac{k}{2^{m}}  \right).
\end{cases}
$$

At interior points of discontinuity a Haar function is put equal to half the sum of its limiting values from the right and from the left, and at the end points of $[0,1]$ to its limiting values from within the interval.

The system $\{\chi_n\}$ was defined by A. Haar in [[#References|[1]]]. It is orthonormal on the interval $[0,1]$. The Fourier series of any continuous function on $[0,1]$ with respect to this system converges uniformly to it. Moreover, if $\omega(\sigma, f)$ is the modulus of continuity of $f$ on $[0,1]$, then the partial sums $S_n(f)$ of order $n$ of the Fourier–Haar series of $f$ satisfy the inequality

<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/h/h046/h046070/h04607018.png" /></td> </tr></table>

The Haar system is a basis in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607022.png" /> is the integral modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607023.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607024.png" />, then (see [[#References|[3]]])

<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/h/h046/h046070/h04607025.png" /></td> </tr></table>

The Haar system is an unconditional basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607027.png" /> (see [[#References|[6]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607028.png" /> is Lebesgue integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607029.png" />, then its Fourier–Haar series converges to it at any of its Lebesgue points; in particular, almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607030.png" />. Here convergence (and absolute convergence) of the Fourier–Haar series at a fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607031.png" /> depends only on the values of the function in any arbitrarily small neighbourhood of this point.

For Fourier–Haar series the following properties differ substantially from each other: a) absolute convergence everywhere; b) absolute convergence almost-everywhere; c) absolute convergence on a set of positive measure; and d) absolute convergence of the series of Fourier coefficients. For trigonometric series all these properties are equivalent.

The properties of the Fourier–Haar coefficients differ sharply from those of the trigonometric Fourier coefficients. For example, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607032.png" /> is continuous on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607033.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607034.png" /> are its Fourier coefficients with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607035.png" />, then the following inequality holds:

<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/h/h046/h046070/h04607036.png" /></td> </tr></table>

which implies that

<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/h/h046/h046070/h04607037.png" /></td> </tr></table>

However, the Fourier–Haar coefficients of continuous functions cannot decrease too rapidly: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607038.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607039.png" /> and if

<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/h/h046/h046070/h04607040.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607042.png" /> (see [[#References|[6]]]).

For functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607044.png" />, the following estimates hold (see [[#References|[3]]]):

<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/h/h046/h046070/h04607045.png" /></td> </tr></table>

<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/h/h046/h046070/h04607046.png" /></td> </tr></table>

<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/h/h046/h046070/h04607047.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607048.png" /> is of bounded variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607050.png" />, then

<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/h/h046/h046070/h04607051.png" /></td> </tr></table>

All these inequalities are sharp in the sense of the order of decrease of their right-hand sides as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607052.png" /> (in the corresponding classes) (see [[#References|[3]]]).

Almost-everywhere unconditionally-converging series of the form

<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/h/h046/h046070/h04607053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

are distinguished by an interesting peculiarity: If a series of the form (*) for any order of its terms converges almost-everywhere on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607054.png" /> of positive Lebesgue measure (the exceptional set of measure 0 may depend on the order of the terms of the series (*)), then this series converges absolutely almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607055.png" />. For series of the form (*) the following criterion holds: For a series (*) to converge almost-everywhere on a measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607056.png" /> it is necessary and sufficient that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607057.png" /> converges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607058.png" /> (see [[#References|[6]]]).

Haar series may serve as representations of measurable functions: For any measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607059.png" /> that is finite almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607060.png" /> there exists a series of the form (*) that converges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607061.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607062.png" />. Here the finiteness of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607063.png" /> is essential: There is no series of the form (*) that converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607064.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046070/h04607065.png" />) on a set of positive Lebesgue measure.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Haar,   "Zur Theorie der orthogonalen Funktionensysteme"  ''Math. Ann.'' , '''69'''  (1910)  pp. 331–371</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Alexits,   "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.L. Ul'yanov,   "On Haar series"  ''Mat. Sb.'' , '''63''' :  3  (1964)  pp. 356–391</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.L. Ul'yanov,   "Absolute and uniform convergence of Fourier series"  ''Math. USSR Sb.'' , '''1''' :  2  (1967)  pp. 169–197  ''Mat. Sb.'' , '''72''' :  2  (1967)  pp. 193–225</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.I. Golubov,   "Series with respect to the Haar system"  ''J. Soviet Math.'' , '''1'''  (1971)  pp. 704–726  ''Itogi. Nauk. Mat. Anal. 1970''  (1971)  pp. 109–143</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.S. Kashin,   A.A. Saakyan,   "Orthogonal series" , Moscow  (1984)  (In Russian)</TD></TR></table>



====Comments====
For a generalization to Banach spaces see [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Singer,   "Bases in Banach spaces" , '''1–2''' , Springer  (1970–1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''1–2''' , Springer  (1977–1979)</TD></TR></table>

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