Haar system

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 [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

$$\sup_{0 \le t \le 1} |f(t) - S_n(t, f)| \le 12 \omega\left(\frac{1}{n}, f\right), \qquad n = 1, 2, \ldots.$$ The Haar system is a basis in the space $L_p[0, 1]$, $1 \le p < \infty$. If $f \in L_p[0, 1]$ and $\omega_p(\delta, f)$ is the integral modulus of continuity of $f$ in the metric of $L_p[0, 1]$, then (see [3])

$$\|f - S_n(f)\|_{L_p[0, 1]} \le 24 \omega_p \left(\frac{1}{n}, f\right), \qquad n = 1, 2, \ldots.$$ The Haar system is an unconditional basis in $L_p[0, 1]$ for $1 < p < \infty$ (see [6]).

If $f$ is Lebesgue integrable on $[0, 1]$, then its Fourier–Haar series converges to it at any of its Lebesgue points; in particular, almost-everywhere on $[0, 1]$. Here convergence (and absolute convergence) of the Fourier–Haar series at a fixed point of $[0, 1]$ 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 $f$ is continuous on the interval $[0, 1]$ and if $a_n(f)$ are its Fourier coefficients with respect to the system $\{\chi_n\}$, then the following inequality holds:

$$|a_n(f)| \le \frac{1}{\sqrt{2n}} \omega \left(\frac{1}{n}, f\right), \qquad n \ge 2,$$ which implies that

$$a_n(f) = o(n^{-1/2}), \qquad n \to\infty.$$ However, the Fourier–Haar coefficients of continuous functions cannot decrease too rapidly: If $f$ is continuous on $[0, 1]$ and if

$$a_n(f) = o(n^{-3/2}),$$ then $f\equiv \text{const}$ on $[0, 1]$ (see [6]).

For functions $f\in L_p[0, 1]$, $1\le p < \infty$, the following estimates hold (see [3]):

$$|a_n(f)| \le n^{(2-p)/2p} \omega_p\left(\frac{1}{n}, f\right), \qquad n = 1, 2, \ldots, $$

$$\left( \sum_{k=2^n + 1}^{2^{n+1}} |a_k(f)|^p \right)^{1/p} \le 8.2^{n(2-p)/2p} \omega_p \left(\frac{1}{2^n}, f\right), \qquad n = 0, 1, \ldots.$$

If $f$ is of bounded variation $V(f)$ on $[0, 1]$, then

$$\sum_{k=2^n+1}^{2^{n+1}} |a_k(f)| \le \left(\frac{3}{2 \sqrt{2^n}}\right) V(f), \qquad n = 0, 1, \ldots.$$ All these inequalities are sharp in the sense of the order of decrease of their right-hand sides as $n\to \infty$ (in the corresponding classes) (see [3]).

Almost-everywhere unconditionally-converging series of the form

$$\sum_{n=1}^\infty a_N \chi_n(t) \tag{*}$$ are distinguished by an interesting peculiarity: If a series of the form (*) for any order of its terms converges almost-everywhere on a set $E\subset[0, 1]$ 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 $[0, 1]$. For series of the form (*) the following criterion holds: For a series (*) to converge almost-everywhere on a measurable set $E\subset [0, 1]$ it is necessary and sufficient that the series $\sum_{n=1}^\infty a_n^2 \chi_n^2(t)$ converges almost-everywhere on $E$ (see [6]).

Haar series may serve as representations of measurable functions: For any measurable function $f$ that is finite almost-everywhere on $[0, 1]$ there exists a series of the form (*) that converges almost-everywhere on $[0, 1]$ to $f$. Here the finiteness of the function $f$ is essential: There is no series of the form (*) that converges to $+\infty$ (or $-\infty$) on a set of positive Lebesgue measure.

References

[1]	A. Haar, "Zur Theorie der orthogonalen Funktionensysteme" Math. Ann. , 69 (1910) pp. 331–371
[2]	G. Alexits, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from Russian)
[3]	P.L. Ul'yanov, "On Haar series" Mat. Sb. , 63 : 3 (1964) pp. 356–391
[4]	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
[5]	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
[6]	B.S. Kashin, A.A. Saakyan, "Orthogonal series" , Moscow (1984) (In Russian)

Comments

For a generalization to Banach spaces see [a1], [a2].

References

[a1]	I.M. Singer, "Bases in Banach spaces" , 1–2 , Springer (1970–1981)
[a2]	J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)