Harmonic analysis on Euclidean spaces Definition

Walsh function

Builds on Rademacher functions Lp space Orthonormal basis

Definition

For $k \ge 1$ , let $r_k$ be the $k$ -th Rademacher function on $[0, 1)$ . Thus $r_k(t) = (-1)^{t_k}$ , where $t_k$ is the $k$ -th binary digit of $t$ , taken from the expansion that does not end in an infinite run of $1$ s. Let $n \ge 0$ be an integer with binary expansion $n = \sum_{k \ge 0} b_k 2^k$ , $b_k \in \set{0, 1}$ . The Walsh function of index $n$ (in Paley order) is $w_n = \prod_{k \ge 0} r_{k+1}^{\,b_k},$ a finite product, with $w_0 = 1$ .

Remarks

The family $(w_n)_{n \ge 0}$ is the Walsh system. It is an orthonormal basis of $L^2([0, 1))$ (see Lp space). Other orderings of the same functions are in use, such as the Walsh order by number of sign changes.

Example: $n = 3 = 1 + 2$ , so $w_3 = r_1 r_2$ . Here $r_1 = 1$ on $[0, \tfrac12)$ and $-1$ on $[\tfrac12, 1)$ , while $r_2 = 1$ on $[0, \tfrac14) \cup [\tfrac12, \tfrac34)$ and $-1$ elsewhere. So $w_3$ is $1, -1, -1, 1$ on the four quarters of $[0, 1)$ , and $\int_0^1 w_3 = 0$ , as orthogonality to $w_0$ requires.

Non-example: the function $\cos(2\pi t)$ is orthogonal to many Walsh functions, but it is not a Walsh function, since it takes values other than $\pm 1$ .

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Lp space concept
Orthonormal basis concept
Rademacher functions concept
Walsh function definition

Tag	`SGXYY92GRS`
URI	`m2://SGXYY92GRS`
Kind	definition
Writing	W0 stub
MSC	42C10
Defines	Walsh function
Uses	Rademacher functions, Lp space, Orthonormal basis

Walsh function

Definition

Remarks

Reading path

Further reading

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