Henstock-Kurzweil type integrals on zero-dimensional groups and its application in Harmonic Analysis

We introduce here a Henstock type integral on compact subsets of a locally compact zero-dimensional abelian group and we use this integral to solve the problem of recovering the coefficients of convergent series with respect to characters of a compact zero dimensional abelian group and to obtain an inversion formula for multiplicative integral transform with a kernel expressed in terms of characters of a locally compact zero dimensional abelian group.