Editing Walsh function (section)

==Definition==

We define the sequence of Walsh functions <math>W_k : [0,1] \rightarrow \{-1,1\}</math>, <math>k\in\mathbb{N}</math> as follows.

For any [[natural number]] ''k'', and [[real number]] <math>x \in [0,1]</math>, let

:<math>k_j</math> be the ''j''th bit in the [[binary representation]] of ''k'', starting with <math>k_0</math> as the least significant bit, and

:<math>x_j</math> be the ''j''th bit in the fractional binary representation of <math>x</math>, starting with <math>x_1</math> as the most significant fractional bit.

Then, by definition

:<math>W_k(x) = (-1)^{\sum_{j=0}^\infty k_jx_{j+1}}</math>

In particular, <math>W_0(x) = 1</math> everywhere on the interval, since all bits of ''k'' are zero.

Notice that <math>W_{2^m}</math> is precisely the [[Rademacher system|Rademacher function]] ''r<sub>m</sub>''.
Thus, the Rademacher system is a subsystem of the Walsh system. Moreover, every Walsh function is a product of Rademacher functions:

:<math>W_k(x) = \prod_{j=0}^\infty r_j(x)^{k_j}</math>