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Biorthogonal systems of Appell polynomials for smooth measures on \({\mathbb R}_{+}\). (English. Ukrainian original) Zbl 1024.46012
Theory Probab. Math. Stat. 59, 1-9 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 1-9 (1998).
Biorthogonal systems of functions \(\{P_{\mu},Q_{\mu}\}\subset L_{2}(\mathcal {F}^{*}, \mu)\), where \(\mathcal {F}^{*}\supset\mathcal{H}\supset\mathcal{F}\) is a rigged real separable Hilbert space and \(\mu\) is a measure on the Borel \(\sigma\)-algebra \(\mathcal{B}(\mathcal{F}^{*})\), are useful in infinite-dimensional analysis.
In this paper, a method for constructing such a biorthogonal system is proposed in the case of a probability measure \(d\mu(s)=p(s)ds,\) \(s\in {\mathbb R}_{+},\) where \(p(s)>0\) is an infinitely differentiable function on \({\mathbb R}_{+},\) under the condition that there exists \(\epsilon>0\) such that \[ \int_{{\mathbb R}_{+}}\exp\{\epsilon s\} d\mu(s)<+\infty. \] In the particular case of \(p(s)=s^{\theta}e^{-s}/\Gamma(\theta+1)\), both systems \(P_{\mu}\) and \(Q_{\mu}\) are constructed in explicit form. These two systems coincide up to normalization with systems of Laguerre polynomials.

46F25 Distributions on infinite-dimensional spaces
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
43A62 Harmonic analysis on hypergroups
60G99 Stochastic processes