Multiplication of Walsh-Paley series and its application (Q1405931)

Let \(\sum\limits_{n=0}^\infty a_nw_n(x)\) be a Walsh series with vanishing coefficients [\textit{B. I. Golubov, A. V. Efimov} and \textit{V. A. Skvortsov}, ``Walsh series and transforms'' (1987; Zbl 0692.42009)]. The author introduces and studies a formal product of the above series and the quickly convergent Fourier-Walsh series. For any sequence \(p_n\downarrow 0\) \((n\to\infty)\), \(p_n\not\equiv0\), and any interval \((\alpha;\beta)\subset [0;1)\) a function \(\lambda(x)\) is constructed which differs from zero on \((\alpha;\beta)\), equals to zero at all points \([0;1)\backslash[\alpha;\beta)\) and whose Fourier-Walsh coefficients \(\hat\lambda(n)=\Bar{\Bar{o}}(p_n)\). Theorems proved for the generalized formal product of the Walsh series are applied to study the properties of kernels of the Walsh zero-series.