Rademacher functions

The first three Rademacher functions

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

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

Rademacher system

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

→ Main article : 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 ${\displaystyle (a_{n})_{n}}$ ${\displaystyle N}$

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

Rademacher averaging

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 .

See also

• Hair wavelet

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.

Web links

• Eric W. Weisstein : Square Wave . In: MathWorld (English).

Individual evidence

1. ^ Eugen Gauß: Walsh functions for engineers and natural scientists . Teubner, Stuttgart 1994, ISBN 3-519-02099-8 (Chapter 3.1).
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 ). ${\displaystyle t=k/2^{n}}$
3. ↑ Peter Karl Huber-Vöckl: Ortho Normal systems, Singular integrals and almost diagonal matrices . (PDF; 1.2 MB) Linz, University, diploma thesis, 2004, p. 9.
4. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Lemma 2.22: Rademacher averaging