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.
The -th Rademacher function is defined by:
- if applies (for a with ).
Alternatively, you can use the -th Rademacher function
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).
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:
The Rademacher functions are then defined as a pair in this context as:
With the above definition it is easier to set up relationships - similar to the trigonometric functions - such as:
The following applies to the function :
and for the function :
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
The Rademacher functions form an orthonormal system of the space of square integrable functions . That is, it applies
where is the Kronecker delta . This orthonormal system is called the Rademacher system, but it is not an orthonormal basis of .
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:
It applies to almost everyone in
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 .
A simple version of this inequality, named after Alexander Yakovlevich Chintschin , in which the Rademacher functions appear, is as follows.
If is a sequence of real numbers, then holds for every natural number
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 :
This formula is called Rademacher averaging. It can be used to estimate norms of the projective tensor product of normalized spaces .
- 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.
- Eric W. Weisstein : Square Wave . In: MathWorld (English).
- ^ Eugen Gauß: Walsh functions for engineers and natural scientists . Teubner, Stuttgart 1994, ISBN 3-519-02099-8 (Chapter 3.1).
- ↑ 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 ).
- ↑ Peter Karl Huber-Vöckl: Ortho Normal systems, Singular integrals and almost diagonal matrices . (PDF; 1.2 MB) Linz, University, diploma thesis, 2004, p. 9.
- ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Lemma 2.22: Rademacher averaging