Fourier transforms and convolutions of generalized measures defined on the dyadic field (Q795265)

The Walsh Fourier transform of a dyadic measure m is defined as \(m{\hat{\;}}(I_ n(x))=(2^{-n})\int^{(2^ n)^-}_{0}w_ y(m)dx. w_ y\) is a Walsh function and \(I_ n(x)\) is the dyadic interval of rank n containing x. The convolution of two dyadic measures m and m' is defined as \[ (m*m')[p/2^ n,\quad p+1^-/2^ n]=\sum^{n}_{k=0}m[(p+k)/2^ n,\quad(p+k)+1^-/2^ n]m'[k/2^ n,\quad(k+1^-)/2^ n]. \] The main result of this paper is the following. Theorem: If dyadic measures m,m' satisfy \(\sum^{\infty}_{p=0}| m[p/2^ n,\quad(p+1^-);2^ n]|<\infty,\) and \(\sup_{x}| m'(I_ n(x))|<\infty\) for each \(n=0,\pm 1,...\), then for each dyadic interval I \[ (m*m'){\hat{\;}}I=\int_{I}\hat f(x)m{\hat{\;}}'(dx), \] \(\hat f\) is a locally integrable function satisfying m{\^{\ }}\(=m_{\hat f}\).