Harmonic analysis on Euclidean spaces Definition

Rademacher functions

Definition

Let $k \ge 1$ be an integer. The $k$ -th Rademacher function $r_k \colon [0, 1) \to \set{1, -1}$ is given by $r_k(t) = \sgn\big(\sin(2^k \pi t)\big)$ where $\sin(2^k \pi t) \ne 0$ . At the finitely many other points of $[0,1)$ , its value is fixed so that $r_k$ is right-continuous. Equivalently, $r_k(t) = (-1)^{t_k}$ , where $t = \sum_{i \ge 1} t_i 2^{-i}$ is the binary expansion of $t$ that does not end in an infinite run of $1$ s. The family $(r_k)_{k \ge 1}$ is the Rademacher system.

Remarks

The Rademacher system is orthonormal in $L^2([0, 1))$ (see Lp space). It is not complete: the product $r_1 r_2$ has norm $1$ and is orthogonal to every $r_k$ . Under Lebesgue measure on $[0, 1)$ , the $r_k$ are independent random variables, each equal to $1$ or $-1$ with probability $\tfrac12$ .

Example: $r_2$ equals $1$ on $[0, \tfrac14)$ and $[\tfrac12, \tfrac34)$ and $-1$ on $[\tfrac14, \tfrac12)$ and $[\tfrac34, 1)$ . Check: $\sin(4 \pi t) > 0$ exactly for $t \in (0, \tfrac14) \cup (\tfrac12, \tfrac34)$ .

Non-example: the function $\sgn(\sin(3 \pi t))$ is a $\pm 1$ square wave on $[0, 1)$ . Its sign changes are at $\tfrac13$ and $\tfrac23$ , not at dyadic points, so it is not a Rademacher function.

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Tag	`BAVME420YM`
URI	`m2://BAVME420YM`
Kind	definition
Writing	W0 stub
MSC	42C10, 60G50
Defines	Rademacher functions
Uses	Lp space, Lebesgue measure

Rademacher functions

Definition

Remarks

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