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

{ t ↦ r n ( t ) = sgn ⁡ ( sin ⁡ 2 n + 1 π t ) ; t ∈ [ 0 , 1 ] , n ∈ N } . {\displaystyle \{t\mapsto r_{n}(t)=\operatorname {sgn} \left(\sin 2^{n+1}\pi t\right);t\in [0,1],n\in \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.

To see that the Rademacher system is an incomplete orthonormal system and not an orthonormal basis, consider the function on the unit interval defined by the following equation:

f ( t ) = 4 | x − 1 2 | − 1 {\displaystyle f(t)=4\left|x-{\frac {1}{2}}\right|-1}

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

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

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