Rademacher system

The orthonormal system $ \{ r _ {k} ( x) \} $ on $ [ 0 , 1 ] $. It was introduced by H. Rademacher [1]. The functions $ r _ {k} ( x) $ are defined by the equations

$$ r _ {k} ( x) = \ \mathop{\rm sign} \sin 2 ^ {k} \pi x ,\ \ x \in [ 0 , 1 ] ,\ \ k = 1 , 2 ,\dots . $$

Another definition of the Rademacher function $ r _ {k} ( x) $ is obtained by considering the binary expansions of numbers in $ [ 0 , 1 ] $: If there is a 0 in the $ k $- th place of the binary expansion of $ x $, then put $ r _ {k} ( x) = 1 $, if there is a 1, then put $ r _ {k} ( x) = - 1 $, if $ x = 0 $ or if $ x $ admits two expansions, then put $ r _ {k} ( x) = 0 $. According to this definition the interval $ [ 0 , 1 ] $ is partitioned into $ 2 ^ {k} $ equal subintervals, in each of which the function $ r _ {k} ( x) $ takes alternately the values $ + 1 $ and $ - 1 $, and at the end points of the subintervals $ r _ {k} ( x) = 0 $.

The system $ \{ r _ {k} ( x) \} $ 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 $ \sum c _ {k} ^ {2} < + \infty $, then the series $ \sum c _ {k} r _ {k} ( x) $ converges almost-everywhere on $ [ 0 , 1 ] $; and the Khinchin–Kolmogorov theorem: If $ \sum c _ {k} ^ {2} = + \infty $, then the series $ \sum c _ {k} r _ {k} ( x) $ diverges almost-everywhere on $ [ 0 , 1 ] $.

Since the Rademacher functions take only the values $ \pm 1 $ on the dyadic irrational points of $ [ 0 , 1 ] $, the consideration of the series $ \sum c _ {k} r _ {n} ( x) $ means that a distribution of $ \pm $ signs is chosen for the terms of the series $ \sum c _ {n} $, depending on $ x $. If $ x = 0 . \alpha _ {1} \alpha _ {2} {} \dots $ is the representation of $ x \in [ 0 , 1 ] $ as an infinite dyadic fraction, then for $ \alpha _ {n} = 0 $ a $ + $ sign is placed in front of $ c _ {n} $ and for $ \alpha _ {n} = 1 $ a $ - $ sign.

In probabilistic terminology the theorem above means that if $ \sum c _ {n} ^ {2} < + \infty $, then $ \sum \pm c _ {n} $ converges for almost-all distributions of signs (converges with probability 1), and if $ \sum c _ {n} ^ {2} = + \infty $, then $ \sum \pm c _ {n} $ 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 $ [ 0 , 1 ] $ the equality

$$ \lim\limits _ {n \rightarrow \infty } \frac{1}{n} \sum _ { k= } 1 ^ { n } r _ {k} ( x) = 0 $$

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 $ \sum _ {k=} 1 ^ \infty r _ {k} ( x) y _ {k} $, where $ x \in [ 0, 1] $ and $ \{ y _ {k} \} _ {k=} 1 ^ \infty $ is a system of vectors in a normed space $ Y $. Of particular interest are the notions of Rademacher type and cotype. A Banach space $ Y $ is said to be of type $ p $, $ 1 \leq p \leq 2 $, if there is a constant $ c $ so that

$$ \int\limits _ { 0 } ^ { 1 } \left \| \sum _ { k= } 1 ^ { n } r _ {k} ( x) y _ {k} \right \| dx \leq \ c \left ( \sum _ { k= } 1 ^ { n } \| y _ {k} \| ^ {p} \right ) ^ {1/p} $$

for all integers $ n $ and all $ \{ y _ {k} \} _ {k=} 1 ^ {n} \subset Y $. The space $ Y $ is said to be of cotype $ q $, $ 2 \leq q < \infty $, if there is a constant $ c $ so that

$$ \left ( \sum _ { k= } 1 ^ { n } \| y _ {k} \| ^ {q} \right ) ^ {1/q} \leq c \int\limits _ { 0 } ^ { 1 } \| \sum r _ {k} ( x ) y _ {k} \| dx $$

for all integers $ n $ and all $ \{ y _ {k} \} _ {k=} 1 ^ {n} \subset Y $. 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)