Editing Walsh function (section)

===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]].