One of the numerical characteristics of a function of several variables that can be regarded as a multi-dimensional analogue of the variation of a function of a single variable. Suppose that a real-valued function
is given on the
-dimensional parallelopipedon
and introduce the notation
Let
be an arbitrary partition of
by hyperplanes
into
-dimensional parallelopipeda, and let
take the values
in an arbitrary way. The Fréchet variation is defined as follows:
If
, then one says that
has bounded (finite) Fréchet variation on
, and the class of all such functions is denoted by
. For
, this class was introduced by M. Fréchet [1] in connection with the investigation of the general form of a bilinear continuous functional
on the space of functions of the form
that are continuous on the square
. He proved that every such functional can be represented in the form
where
,
.
Analogues of many of the classical criteria for the convergence of Fourier series are valid for
-periodic functions in the class
(
, see [2]). For example, if
,
then the rectangular partial sums of the Fourier series of
converge at every point
to the number
where the summation is taken over all the
possible combinations of the signs
. Here, if the function is continuous, the convergence is uniform (an analogue of the Jordan criterion).
References
[1] | M. Fréchet, "Sur les fonctionelles bilinéaires" Trans. Amer. Math. Soc. , 16 : 3 (1915) pp. 215–234 |
[2] | M. Morse, W. Transue, "The Fréchet variation and the convergence of multiple Fourier series" Proc. Nat. Acad. Sci. USA , 35 : 7 (1949) pp. 395–399 |
How to Cite This Entry:
Fréchet variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_variation&oldid=14364
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article