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In mathematics, in particular in
Rademacher system
in the '' Encyclopedia of Mathematics'' Functional analysis {{Mathanalysis-stub
In mathematics, in particular in functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Rademacher system, named after Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892 – 7 February 1969) was a German-born American mathematician, known for work in mathematical analysis and number theory.
Biography
Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göt ...
, is an incomplete orthogonal system of functions on the unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
of the following form:
:
The Rademacher system is stochastically independent, and is closely related to the Walsh
Walsh may refer to:
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* Walsh (surname), including a list of people and fictional characters
Places Australia
* Mount Walsh, Mount Walsh National Park
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* Fort Walsh, one of the first Royal Canadian Mounted ...
system. Specifically, the Walsh system can be constructed as a product of Rademacher functions.
To see that the Rademacher system is an incomplete orthogonal system and not an orthonormal basis, consider the function on the unit interval defined by the following equation:
This function is orthogonal to all the functions in the Rademacher system, yet is nonzero.
References
* * * *External links
Rademacher system
in the '' Encyclopedia of Mathematics'' Functional analysis {{Mathanalysis-stub