Rademacher function

The Rademacher, named after Hans Rademacher, are defined for each natural number on the ( semi-open ) unit interval [0,1) functions that take only the values ​​-1 and 1.

Definition

The - te Rademacherfunktion is defined as follows:

Alternatively, the - te Rademacherfunktion by

Define. This definition is equivalent to the first definition for all numbers that are not of the form. When has this form, it is and therefore the sign ( sgn ) disappears. The difference, however, concerns only finitely many for each and therefore plays eg in function spaces no role ( because the functions can here be changed to null sets ).

In the literature, the Rademacherfunktionenen occasionally continued periodically outside of the base interval and the definition of Rademacher is with respect to the Walsh - Kaczmarz functions " Walsh sine " and " cosine Walsh " as:

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

With the above setting will be easier to cover, similar to the trigonometric functions form such as:

Examples

Therefore applies to the function:

And for the function:

General orders the -th Rademacher function of a number in the unit interval to a -1 if the second digit in the binary representation of a 1, and a 1 if this number is 0. For example, applies

And

Rademachersystem

The Rademacher form an orthonormal system of the space of square integrable functions. That is, it is

Where the Kronecker delta is. This orthonormal system named Rademachersystem, but there is no orthonormal basis of.

Normal numbers

The number is called simply normal to the base 2 (see also normal number ), if the two digits 0 and 1 occur frequently equal in their binary representation. The fact that almost all numbers are just normal, can be described using the Rademacher as:

It applies to almost all t in [ 0,1)

If one interprets the binary representation of each of the numbers in the unit interval as an infinite sequence of coin tosses ( Bernoulli process ), this is just the statement of the strong law of large numbers.

Khinchin 's inequality

A simple version of this inequality, which is named after Alexander Yakovlevich Khinchin and in the the Rademacher occur is as follows.

Is a sequence of real numbers, then for every natural number