The Rademacher functions , named after Hans Rademacher , are functions defined for every natural number on the (half-open) unit interval [0,1), which only take the values ​​−1 and 1. ${\ displaystyle n}$

## definition

The -th Rademacher function is defined by: ${\ displaystyle n}$

${\ displaystyle r_ {n} (t): = (- 1) ^ {k} \,}$if applies (for a with ).${\ displaystyle {\ frac {k} {2 ^ {n}}} \ leq t <{\ frac {k + 1} {2 ^ {n}}}}$${\ displaystyle k}$${\ displaystyle 0 \ leq k <2 ^ {n} -1}$

Alternatively, you can use the -th Rademacher function ${\ displaystyle n}$

${\ displaystyle r_ {n} (t): = \ operatorname {sgn} {\ big (} \ sin {\ big (} 2 ^ {n} \ pi t {\ big)} {\ big)}}$

define. This definition is equivalent to the first definition for all numbers that are not of the form . If this has, then is and therefore the sign ( so-called ) also disappears . However, the difference only affects a finite number of each and therefore plays e.g. B. in function spaces like no role (since here the functions can be changed to zero sets at will). ${\ displaystyle t}$${\ displaystyle k / 2 ^ {n} \,}$${\ displaystyle t}$${\ displaystyle \ sin \ left (2 ^ {n} \ pi t \ right) = 0}$${\ displaystyle n}$${\ displaystyle t}$${\ displaystyle L ^ {2} ([0,1])}$

In the literature, the Rademacher functions are occasionally continued periodically outside the base interval and the Rademacher functions are defined with reference to the Walsh-Kaczmarz functions "Walsh sine" and "Walsh cosine" as: ${\ displaystyle \ operatorname {sir}}$${\ displaystyle \ operatorname {cor}}$

${\ displaystyle \ operatorname {sir} (x): = (- 1) ^ {\ lfloor 2x \ rfloor} = \ operatorname {sign} (\ sin (2 \ pi x))}$
${\ displaystyle \ operatorname {cor} (x): = (- 1) ^ {\ lfloor 2x + {\ frac {1} {2}} \ rfloor} = \ operatorname {sign} (\ cos (2 \ pi x) )}$

The Rademacher functions are then defined as a pair in this context as:

${\ displaystyle \ operatorname {sir} (2 ^ {n} x)}$
${\ displaystyle \ operatorname {cor} (2 ^ {n} x)}$

With the above definition it is easier to set up relationships - similar to the trigonometric functions - such as:

${\ displaystyle \ operatorname {sir} (x) \ operatorname {cor} (x) = \ operatorname {sir} (2x)}$

## Examples

The following applies to the function : ${\ displaystyle r_ {1} (t) \,}$

${\ displaystyle r_ {1} (t) = {\ begin {cases} 1 \ quad & 0 \ leq t <1/2, \\ - 1 & 1/2 \ leq t <1, \ end {cases}}}$

and for the function : ${\ displaystyle r_ {2} (t) \,}$

${\ displaystyle r_ {2} (t) = {\ begin {cases} 1 \ quad & 0 \ leq t <1/4, \\ - 1 & 1/4 \ leq t <1/2, \\ 1 \ quad & 1 / 2 \ leq t <3/4, \\ - 1 & 3/4 \ leq t <1. ​​\\\ end {cases}}}$

In general, the -th Rademacher function assigns a number in the unit interval a -1 if the -th digit in the binary representation of is a 1, and a 1 if this digit is 0. For example ${\ displaystyle n}$${\ displaystyle t}$${\ displaystyle n}$${\ displaystyle t}$

${\ displaystyle r_ {1} (0 {,} 375) = r_ {1} (0 {,} {\ color {red} 0} 11_ {2}) = 1}$

and

${\ displaystyle r_ {2} (0 {,} 375) = r_ {2} (0,0 {\ color {red} 1} 1_ {2}) = - 1}$.

The Rademacher functions form an orthonormal system of the space of square integrable functions . That is, it applies ${\ displaystyle L ^ {2} ([0,1])}$

${\ displaystyle \ int _ {0} ^ {1} r_ {n} (x) r_ {m} (x) \ mathrm {d} x = \ delta _ {mn}}$,

where is the Kronecker delta . This orthonormal system is called the Rademacher system, but it is not an orthonormal basis of . ${\ displaystyle \ delta _ {mn}}$${\ displaystyle L ^ {2} ([0,1])}$

## Normal numbers

The number is simply called normal to base 2 (see also normal number ) if the two digits 0 and 1 appear equally often in their binary representation. The fact that almost all numbers are simply normal can be described with the help of the Rademacher functions: ${\ displaystyle t \ in [0,1)}$

It applies to almost everyone in${\ displaystyle t}$${\ displaystyle [0,1)}$

${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {r_ {1} (t) + \ cdots + r_ {n} (t)} {n}} = 0.}$

If one interprets the binary representation of each of the numbers in the unit interval as an infinite sequence of coin tosses ( Bernoulli process with ), then that is precisely the statement of the strong law of large numbers . ${\ displaystyle p = 1/2}$

## Chinchin inequality

A simple version of this inequality, named after Alexander Yakovlevich Chintschin , in which the Rademacher functions appear, is as follows. ${\ displaystyle r_ {n} (t) \,}$

If is a sequence of real numbers, then holds for every natural number${\ displaystyle (a_ {n}) _ {n}}$${\ displaystyle N}$

${\ displaystyle \ int _ {0} ^ {1} \ left | \ sum _ {n = 1} ^ {N} a_ {n} r_ {n} (t) \ right | \ mathrm {d} t \ geq {\ frac {1} {\ sqrt {2}}} \ left (\ sum _ {n = 1} ^ {N} a_ {n} ^ {2} \ right) ^ {1/2}.}$

If and are vector spaces, the Rademacher functions can be used to find alternative representations of elements from the tensor product . It applies to everyone and : ${\ displaystyle E}$${\ displaystyle F}$ ${\ displaystyle E \ otimes F}$${\ displaystyle x_ {1}, \ ldots, x_ {n} \ in E}$${\ displaystyle y_ {1}, \ ldots, y_ {n} \ in F}$

${\ displaystyle \ sum _ {i = 1} ^ {n} x_ {i} \ otimes y_ {i} = \ int _ {0} ^ {1} \ left (\ sum _ {i = 1} ^ {n } r_ {i} (t) x_ {i} \ right) \ otimes \ left (\ sum _ {i = 1} ^ {n} r_ {i} (t) y_ {i} \ right) \, \ mathrm {d} t}$.

This formula is called Rademacher averaging. It can be used to estimate norms of the projective tensor product of normalized spaces .

## literature

• Hans Rademacher: Some theorems about series of general orthogonal functions . In: Mathematical Annals . tape 87 , no. 1/2 , 1922, ISSN  0025-5831 , p. 112-138 ( online ).
• Mark Kac : Statistical independence in probability, analysis and number theory . Ed .: Mathematical Association of America (=  The Carus Mathematical Monographs . Volume 12 ). Ithaca NY 1959, ISBN 0-88385-012-5 (Chapters 1 and 2: Application to coin toss ).
• Stefan Kaczmarz , Hugo Steinhaus : Theory of the orthogonal series (=  monograph Matematyczne . Volume 6 ). Z Subwencji Funduszu Kultury Narodowej, 1935, ISSN  0077-0507 ( matwbn.icm.edu.pl - especially Chapter 4).
• Donald E. Knuth : The Art of Computer Programming. Volume 4, A: Combinatorial algorithms. Part 1. Addison-Wesley, Upper Saddle River NJ u. a. 2011, ISBN 978-0-201-03804-0 , especially pp. 287-288.

2. This description is ambiguous for numbers of the form (which are also called dyadic rational numbers). These numbers have two binary representations (example: 1/2 = 0.1 2 = 0.0111… 2 ).${\ displaystyle t = k / 2 ^ {n}}$