Recovery of the coefficients of multiple Haar and Walsh series. (English) Zbl 1168.42014

Plotnikov, Mikhail G.

Real Anal. Exch. 33(2007-2008), No. 2, 291-308 (2008).

The problem of recovery of the coefficients of an orthogonal series from its sum is considered. It is known that in several situations this requires an integration process more general than the classical Lebesgue integration. In this paper the author defines some multidimensional Perron type integrals in terms of the dyadic base of differentiation. These integrals are used to solve the coefficient problem, via formal integration and generalized Fourier formulae, for some multiple Haar and Walsh series. This class of series includes in particular those convergent \(\rho\)-regularly everywhere except some countable set. The author analyzes the two-dimensional case and shows that some properties of rectangularly convergent double Haar and Walsh series do not hold for the \(\rho\)-regular convergence.

Reviewer: Andrei Martínez Finkelshtein (Almeria)

MSC:

42C25	Uniqueness and localization for orthogonal series
26A39	Denjoy and Perron integrals, other special integrals
42B05	Fourier series and coefficients in several variables
42B08	Summability in several variables

Keywords:

dyadic group; multiple Haar series; multiple Walsh series; dyadic derivative; Perron type integral; Fourier formulae; sets of uniqueness