Editing Walsh function (section)

==Properties==

The Walsh system <math>\{W_k\}, k \in \mathbb{N}_0</math> is an [[abelian group|abelian]] multiplicative [[discrete group]] [[isomorphic]] to <math>\coprod_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}</math>, the [[Pontryagin duality|Pontryagin dual]] of the [[Cantor cube|Cantor group]] <math>\prod_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}</math>. Its [[identity element|identity]] is <math>W_0</math>, and every element is of [[order (group theory)|order]] two (that is, self-inverse).

The Walsh system is an orthonormal basis of the Hilbert space <math>L^2[0,1]</math>. Orthonormality means

:<math>\int_0^1 W_k(x)W_l(x) dx = \delta_{kl}</math>,

and being a basis means that if, for every <math>f \in L^2[0,1]</math>, we set <math>f_k = \int_0^1 f(x)W_k(x)dx</math> then

:<math>\int_0^1 ( f(x) - \sum_{k=0}^N f_k W_k(x) )^2 dx \;\ \xrightarrow[N\rightarrow\infty]{}\;\ 0</math>

It turns out that for every <math>f \in L^2[0,1]</math>, the [[series (mathematics)|series]] <math>\sum_{k=0}^\infty f_k W_k(x)</math> [[convergent series|converges]] to <math>f(x)</math> for almost every <math>x \in [0,1]</math>.

The Walsh system (in Walsh-Paley numeration) forms a [[Schauder basis]] in <math>L^p[0,1]</math>, &nbsp; <math>1 < p < \infty</math>. Note that, unlike the [[Haar wavelet|Haar system]], and like the trigonometric system, this basis is not [[Schauder basis|unconditional]], nor is the system a Schauder basis in <math>L^1[0,1]</math>.