Integral operators and dual orthogonal systems on a half-line

摘要

A generalized Laplace transformation μ→W_(θ,μ), on the set of probability measures on R_+, is introduced. The kernel of the transformation is chosen to be (w-tilde)_θ(zs) = _0F_1(θ + 1; zs) (_0F_1 is the hypergeometric function, θ > -1, s ∈R_+, z ∈ C). A family of measures M_θ = {μ : W_(θ,μ) ∈ L}, where L stands for the set of Laguerre entire functions, is studied. The set L consists of polynomials with real nonpositive zeros only, as well as of their uniform limits on compact subsets of C. The set M_θ contains, among others, the Euler measure d_(Υθ) = (s~θ/Γ(θ+1)) exp (-s) ds and the Dirac measures δ_x, x ∈ R_+, which play a peculiar role in the Urbanik algebras defined by the transformation μ → W_(θ,μ). A sufficient condition for the measures d_μ(s) = C_θs~θ exp (-Φ(s)) ds to belong to M_θ, is given. For μ ∈ M_θ, integral operators with the kernels K_θ~μ(z,s) = w_θ(zs)/W_(θ,μ)(z), acting in the real Hilbert spaces L~2(R_+,dμ), are studied. In particular, dual orthogonal systems of Appell polynomials are constructed.