We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional OrnsteinUhlenbeck processes is derived.

In the present article, using generalized Bessel-Maitland transform, Laplace transform and other known transforms, authors obtain new Parseval-Goldstein type relations. Using these relations, some integrals involving Fox-Wright functions are evaluated. Illustrative examples also given.

We study the determinant line bundle over moduli space of stable bundles on abelian surfaces. We evaluate their analytic torsions. We extend Mukai's version of the Parseval Theorem to L 2 metrics on cohomology groups. We prove that the Mukai transform preserves the determinant line bundle as a hermitian line bundle. This is done by induction via the natural boundary of the moduli spaces.