One of the classical
orthonormal systems of functions.
The
Haar functions
of this system are defined on the interval
as follows:
if
,
,
then
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
to its limiting values from within the interval.
The system
was defined by
A. Haar
in
[1].
It is orthonormal on the interval
.
The Fourier series of any continuous function on
with respect to this system converges uniformly to it. Moreover, if
is the
modulus of continuity
of
on
,
then the partial sums
of order
of the
Fourier–Haar series
of
satisfy the inequality
The Haar system is a basis in the space
,
.
If
and
is the
integral modulus of continuity
of
in the metric of
,
then (see
[3])
The Haar system is an
unconditional basis
in
for
(see
[6]).
If
is Lebesgue integrable on
,
then its Fourier–Haar series converges to it at any of its
Lebesgue points;
in particular, almost-everywhere on
.
Here convergence (and absolute convergence) of the
Fourier–Haar series at a fixed point of
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
is continuous on the interval
and if
are its Fourier coefficients with respect to the system
,
then the following inequality holds:
which implies that
However, the Fourier–Haar coefficients of continuous
functions cannot decrease too rapidly: If
is continuous on
and if
then
on
(see
[6]).
For functions
,
,
the following estimates hold (see
[3]):
If
is of bounded variation
on
,
then
All these inequalities are sharp in the sense of
the order of decrease of their right-hand sides as
(in the corresponding classes) (see
[3]).
Almost-everywhere unconditionally-converging series of the form
are distinguished by an interesting peculiarity: If a series of the form
(*)
for any order of its terms converges almost-everywhere on a set
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
.
For series of the form
(*)
the following criterion holds: For
a series
(*)
to converge almost-everywhere on a measurable set
it is necessary and sufficient that the series
converges almost-everywhere on
(see
[6]).
Haar series may serve as representations of
measurable functions: For any measurable function
that is finite almost-everywhere on
there exists a series of the form
(*)
that converges almost-everywhere on
to
.
Here the finiteness of the function
is essential: There is no series of the form
(*)
that converges to
(or
)
on a set of positive Lebesgue measure.