The Parseval equality for multiple Fourier-Stieltjes series with respect to the Haar system

Abstract

Let a function f : $$\Pi ^{ * ^m } $$ → ℂ be Lebesgue integrable on $$\Pi ^{ * ^m } $$ and Riemann-Stieltjes integrable with respect to a function G : $$\Pi ^{ * ^m } $$ → ℂ on $$\Pi ^{ * ^m } $$ . Then the Parseval equality holds, where (k) = (f, χk) = (L) f(x)χk(x) dx and $$\widehat{dG}$$ (k) = χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Π m → ℂ is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Π m , are Fourier-Lebesgue coefficients of the function f x(t) = f(x ⊕ t), where ⊕ is the group addition, then the Parseval equality holds for almost all x ∈ $$\Pi ^{ * ^m } $$ in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).