Rademacher system

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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:

$\{t\mapsto r_n(t)=\mathrm{sgn}(\sin 2^{n+1}\pi t);t\in [0,1],n\in \mathbb{\mathbb{N}}\}.$

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.

References

• Rademacher, Hans (1922). "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen". Math. Ann. 87 (1): 112–138. doi:10.1007/BF01458040. S2CID 120708120.
• "Orthogonal system", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Heil, Christopher E. (1997). "A basis theory primer" (PDF).
• Curbera, Guillermo P. (2009). "How Summable are Rademacher Series?". Vector Measures, Integration and Related Topics. Basel: Birkhäuser Basel. pp. 135–148. doi:10.1007/978-3-0346-0211-2_13. ISBN 978-3-0346-0210-5.

External links

• Rademacher system in the Encyclopedia of Mathematics