Spectral measure of commutative Jacobi field equipped with multiplication structure

Abstract

The article investigates properties of the spectral measure of the Jacobi field constructed over an abstract Hilbert rigging $H_-\supset H\supset L\supset H_+.$ Here $L$ is a real commutative Banach algebra that is dense in $H.$ It is shown that with certain restrictions, the Fourier transform of the spectral measure can be found in a similar way as it was done for the case of the Poisson field with the zero Hilbert space $L^2(\Delta,d u).$ Here $\Delta$ is a Hausdorff compact space and $ u$ is a probability measure defined on the Borel $\sigma$-algebra of subsets of $\Delta.$ The article contains a formula for the Fourier transform of a spectral measure of the Jacobi field that is constructed over the above-mentioned abstract rigging.