Editing Walsh function (section)

==Generalizations==

===Walsh-Ferleger systems===

Let <math>\mathbb{D} = \prod_{n=1}^\infty \mathbb{Z}/2\mathbb{Z}</math> be the [[compact group|compact]] [[Cantor cube|Cantor group]] endowed with [[Haar measure]] and let <math> \hat {\mathbb D} = \coprod_{n=1}^\infty \mathbb Z / 2\mathbb Z </math> be its discrete group of [[Character (mathematics)|characters]]. Elements of <math> \hat {\mathbb D} </math> are readily identified with Walsh functions. Of course, the characters are defined on <math> \mathbb D </math> while Walsh functions are defined on the unit interval, but since there exists a [[Standard probability space|modulo zero isomorphism]] between these [[measure space]]s, measurable functions on them are identified via [[isometry]].

Then basic [[representation theory]] suggests the following broad generalization of the concept of '''Walsh system'''.

For an arbitrary [[Banach space]] <math>(X, ||\cdot||)</math> let <math>\{ R_t \}_{t \in \mathbb D} \subset \operatorname{Aut}X</math> be a [[Strong operator topology|strongly continuous]], uniformly bounded [[Group action#Notable properties of actions|faithful]] [[group action|action]] of <math>\mathbb D</math> on ''X''. For every <math>\gamma \in \hat{\mathbb D}</math>, consider its [[eigenspace]] <math> X_\gamma = \{x \in X : R_t x = \gamma(t)x \}</math>. Then  ''X'' is the closed linear span of the eigenspaces: <math>X = \overline{\operatorname{Span}}(X_\gamma, \gamma \in \hat {\mathbb D})</math>. Assume that every eigenspace is one-[[dimension (vector space)|dimensional]] and pick an element <math>w_\gamma \in X_\gamma</math> such that <math>\|w_\gamma\| = 1</math>. Then the system <math>\{w_\gamma\}_{\gamma \in \hat {\mathbb D}}</math>, or the same system in the Walsh-Paley numeration of the characters <math>\{w_k\}_{k \in {\mathbb N}_0}</math> is called generalized Walsh system associated with action <math>\{ R_t \}_{t \in \mathbb D}</math>. Classical Walsh system becomes a special case, namely, for

:<math>R_t: x = \sum_{j=1}^\infty x_j2^{-j} \mapsto \sum_{j=1}^\infty (x_j \oplus t_j)2^{-j}</math>

where <math>\oplus</math> is addition [[modular arithmetic|modulo]] 2.

In the early 1990s, Serge Ferleger and Fyodor Sukochev showed that in a broad class of Banach spaces (so called ''UMD'' spaces<ref>{{harvnb|Pisier|2011}}.</ref>) generalized Walsh systems have many properties similar to the classical one: they form a Schauder basis<ref>{{harvnb|Sukochev|Ferleger|1995}}.</ref> and a uniform finite-dimensional decomposition<ref>{{harvnb|Ferleger|Sukochev|1996}}.</ref> in the space, have property of random unconditional convergence.<ref>{{harvnb|Ferleger|1998}}.</ref>
One important example of generalized Walsh system is Fermion Walsh system in non-commutative ''L''<sup>p</sup> spaces associated with [[hyperfinite type II factor]].

===Fermion Walsh system===

The '''Fermion Walsh system''' is a non-commutative, or "quantum" analog of the classical Walsh system. Unlike the latter, it consists of operators, not functions. Nevertheless, both systems share many important properties, e.g., both form an orthonormal basis in corresponding Hilbert space, or [[Schauder basis]] in corresponding symmetric spaces. Elements of the Fermion Walsh system are called ''Walsh operators''.

The term ''Fermion'' in the name of the system is explained by the fact that the enveloping operator space, the so-called [[hyperfinite type II factor]] <math> \mathcal R</math>, may be viewed as the space of ''observables'' of the system of countably infinite number of distinct [[Spin (physics)|spin]] <math>1/2</math> [[fermion]]s. Each [[Rademacher function|Rademacher]] operator acts on one particular fermion coordinate only, and there it is a [[Pauli matrices|Pauli matrix]]. It may be identified with the observable measuring spin component of that fermion along one of the axes <math> \{x,y,z\}</math> in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.

===Vilenkin system===

Fix a sequence <math>\alpha = (\alpha_1,\alpha_2,...)</math> of [[integer]]s with <math>\alpha_k \geq 2, k=1,2,\dots</math> and let <math>\mathbb{G} = \mathbb{G}_\alpha = \prod_{n=1}^\infty \mathbb{Z}/\alpha_k\mathbb{Z}</math> endowed with the [[product topology]] and the normalized Haar measure. Define <math>A_0 = 1</math> and <math>A_k = \alpha_1 \alpha_2 \dots \alpha_{k-1}</math>. Each <math>x \in \mathbb G</math> can be associated with the real number

:<math> \left|x\right| = \sum_{k=1}^{\infty} \frac{x_k}{A_{k}} \in \left[0,1\right].</math>

This correspondence is a module zero isomorphism between <math>\mathbb G</math> and the unit interval. It also defines a norm which generates the [[topological space|topology]] of <math>\mathbb G</math>. For <math>k=1,2,\dots</math>, let <math>\rho_k: \mathbb{G}\to\mathbb{C}</math> where

:<math>\rho_k(x) = \exp\left(i\frac{2 \pi x_k}{\alpha_k}\right) = \cos\left(\frac{2 \pi x_k}{\alpha_k}\right) + i \sin\left(\frac{2 \pi x_k}{\alpha_k}\right).</math>

The set <math>\{\rho_k\}</math> is called ''generalized Rademacher system''. The Vilenkin system is the [[group (mathematics)|group]] <math>\hat{\mathbb G} = \coprod_{n=1}^\infty \mathbb{Z}/\alpha_k\mathbb{Z}</math> of ([[complex number|complex]]-valued) characters of <math>\mathbb G</math>, which are all finite products of <math>\{\rho_k\}</math>. For each non-negative integer <math>n</math> there is a unique sequence <math>n_0, n_1, \dots</math> such that <math>0 \leq n_k < \alpha_{k+1}, k=0,1,2,\dots</math> and

:<math>n = \sum_{k=0}^{\infty} n_k A_k.</math>

Then <math>\hat{\mathbb G} = {\chi_n | n=0,1,\dots}</math> where

:<math>\chi_n = \sum_{k=0}^{\infty} \rho_{k+1}^{n_k}.</math>

In particular, if <math>\alpha_k = 2, k=1,2...</math>, then <math>\mathbb G</math> is the Cantor group and <math>\hat{\mathbb G} = \left\{\chi_n | n=0,1,\dots\right\}</math> is the (real-valued) Walsh-Paley system.

The Vilenkin system is a complete orthonormal system on <math> \mathbb G </math> and forms a [[Schauder basis]] in <math>L^p(\mathbb{G}, \mathbb{C})</math>,&nbsp; <math>1 < p < \infty</math>.<ref>{{harvnb|Young|1976}}</ref>

===Nonlinear Phase Extensions===
Nonlinear phase extensions of discrete Walsh-[[Hadamard transform]] were developed. It was shown that the nonlinear phase basis functions with improved cross-correlation properties significantly outperform the traditional Walsh codes in code division multiple access (CDMA) communications.<ref>A.N. Akansu and R. Poluri,  [http://web.njit.edu/~akansu/PAPERS/Akansu-Poluri-WALSH-LIKE2007.pdf "Walsh-Like Nonlinear Phase Orthogonal Codes for Direct Sequence CDMA Communications,"] IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3800–3806, July 2007.</ref>