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. Viewed as a sequence of random variables on the unit interval with the Lebesgue measure, the law of each r n {\displaystyle r_{n}} is a Rademacher distribution, and the r n {\displaystyle r_{n}} 's are independent of each other. From Khintchine's inequality, it follows that the closed linear span of the Rademacher functions in L p {\displaystyle L^{p}} is the same as the closed linear span in L 2 {\displaystyle L^{2}} .
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