Rademacher system

File:Rademacherfunktionen.png

In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form:

: $\{ t \mapsto r_{n}(t)=\sgn \left( \sin 2^{n+1} \pi t \right) ; t \in [0,1], n \in \N \}.$

The Rademacher system is stochastically independent, and is closely related to the Walsh 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:

$f(t) = 4 \left|x-\frac 12\right| - 1$

This function is orthogonal to all the functions in the Rademacher system, yet is nonzero.

References

{{cite journal |last1=Rademacher |first1=Hans |title=Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen |journal=Math. Ann. |date=1922 |volume=87 |issue=1 |pages=112–138 |doi=10.1007/BF01458040|s2cid=120708120 }}
{{SpringerEOM| title=Orthogonal system}}
{{Cite web| last=Heil | first=Christopher E. | title=A basis theory primer | url=http://www.math.gatech.edu/~heil/papers/bases.pdf | date=1997 }}
{{cite book | last=Curbera | first=Guillermo P. | title=Vector Measures, Integration and Related Topics | chapter=How Summable are Rademacher Series? | publisher=Birkhäuser Basel | publication-place=Basel | year=2009 | pages=135–148 | isbn=978-3-0346-0210-5 | doi=10.1007/978-3-0346-0211-2_13 }}

External links

[https://encyclopediaofmath.org/wiki/Rademacher_system Rademacher system] in the Encyclopedia of Mathematics

Category:Functional analysis