Rademacher system
The orthonormal system on
. It was introduced by H. Rademacher [1]. The functions
are defined by the equations
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Another definition of the Rademacher function is obtained by considering the binary expansions of numbers in
: If there is a 0 in the
-th place of the binary expansion of
, then put
, if there is a 1, then put
, if
or if
admits two expansions, then put
. According to this definition the interval
is partitioned into
equal subintervals, in each of which the function
takes alternately the values
and
, and at the end points of the subintervals
.
The system is a typical example of a system of stochastically-independent functions and has applications both in probability theory and in the theory of orthogonal series.
One of the major properties of the Rademacher series is shown by Rademacher's theorem: If , then the series
converges almost-everywhere on
; and the Khinchin–Kolmogorov theorem: If
, then the series
diverges almost-everywhere on
.
Since the Rademacher functions take only the values on the dyadic irrational points of
, the consideration of the series
means that a distribution of
signs is chosen for the terms of the series
, depending on
. If
is the representation of
as an infinite dyadic fraction, then for
a
sign is placed in front of
and for
a
sign.
In probabilistic terminology the theorem above means that if , then
converges for almost-all distributions of signs (converges with probability 1), and if
, then
diverges for almost-all distributions of signs (diverges with probability 1).
Conversely, a number of theorems in probability theory can be formulated in terms of Rademacher functions. For example, Cantelli's theorem (that for the game of "heads and tails" with stake 1, the average gain tends to zero with probability 1) means that almost-everywhere on the equality
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is satisfied.
References
[1] | H. Rademacher, "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen" Math. Ann. , 87 (1922) pp. 112–138 |
[2] | S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) |
[3] | G. Alexits, "Convergence problems of orthogonal series" , Pergamon (1961) (Translated from German) |
[4] | M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1959) |
Comments
In the study of normed spaces as well as in probability theory, an important role is played by the behaviour of series of the form , where
and
is a system of vectors in a normed space
. Of particular interest are the notions of Rademacher type and cotype. A Banach space
is said to be of type
,
, if there is a constant
so that
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for all integers and all
. The space
is said to be of cotype
,
, if there is a constant
so that
![]() |
for all integers and all
. The spaces which are of type 2 and cotype 2 are exactly those which are isomorphic to a Hilbert space.
References
[a1] | J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , II. Function spaces , Springer (1979) |
Rademacher system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rademacher_system&oldid=48409