View source for Rademacher system

The [[Orthonormal system|orthonormal system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770301.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770302.png" />. It was introduced by H. Rademacher [[#References|[1]]]. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770303.png" /> are defined by the equations

<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/r/r077/r077030/r0770304.png" /></td> </tr></table>

Another definition of the Rademacher function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770305.png" /> is obtained by considering the binary expansions of numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770306.png" />: If there is a 0 in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770307.png" />-th place of the binary expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770308.png" />, then put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r0770309.png" />, if there is a 1, then put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703010.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703011.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703012.png" /> admits two expansions, then put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703013.png" />. According to this definition the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703014.png" /> is partitioned into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703015.png" /> equal subintervals, in each of which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703016.png" /> takes alternately the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703018.png" />, and at the end points of the subintervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703019.png" />.

The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703021.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703022.png" /> converges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703023.png" />; and the Khinchin–Kolmogorov theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703024.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703025.png" /> diverges almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703026.png" />.

Since the Rademacher functions take only the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703027.png" /> on the dyadic irrational points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703028.png" />, the consideration of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703029.png" /> means that a distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703030.png" /> signs is chosen for the terms of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703031.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703033.png" /> is the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703034.png" /> as an infinite dyadic fraction, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703035.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703036.png" /> sign is placed in front of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703037.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703038.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703039.png" /> sign.

In probabilistic terminology the theorem above means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703041.png" /> converges for almost-all distributions of signs (converges with probability 1), and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703043.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703044.png" /> the equality

<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/r/r077/r077030/r07703045.png" /></td> </tr></table>

is satisfied.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Rademacher,   "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen"  ''Math. Ann.'' , '''87'''  (1922)  pp. 112–138</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Alexits,   "Convergence problems of orthogonal series" , Pergamon  (1961)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Kac,   "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer.  (1959)</TD></TR></table>



====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703048.png" /> is a system of vectors in a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703049.png" />. Of particular interest are the notions of Rademacher type and cotype. A Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703050.png" /> is said to be of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703052.png" />, if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703053.png" /> so 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/r/r077/r077030/r07703054.png" /></td> </tr></table>

for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703055.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703056.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703057.png" /> is said to be of cotype <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703059.png" />, if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703060.png" /> so 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/r/r077/r077030/r07703061.png" /></td> </tr></table>

for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703062.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077030/r07703063.png" />. The spaces which are of type 2 and cotype 2 are exactly those which are isomorphic to a Hilbert space.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Lindenstrauss,   L. Tzafriri,   "Classical Banach spaces" , '''II. Function spaces''' , Springer  (1979)</TD></TR></table>