An infinite dimensional umbral calculus

Finkelshtein, Dmitri; Kondratiev, Yuri; Lytvynov, Eugene; Oliveira, Maria João

doi:10.1016/j.jfa.2019.03.006

Cited by 5 publications

(25 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is a formal series in powers of z ∈ C with coefficients from Φ. See [11] for further details on such formal power series.…”

Section: Holomorphic Mappings On Complex Nuclear and Co-nuclear Spacesmentioning

confidence: 99%

“…Here P (k) (ω) P (n−k) (ζ) ∈ Φ n is the symmetrization of P (k) (ω) ⊗ P (n−k) (ζ) ∈ Φ ⊗n . By [11,Theorem 4.1], a sequence of monic polynomials, (P (n) ) ∞ n=0 , is of binomial type if and only if it has the generating function…”

Section: Sheffer Sequences On φmentioning

confidence: 99%

“…Let now (S (n) ) ∞ n=0 be a Sheffer sequence and let (P (n) ) ∞ n=0 be its basic sequence. We will use the lemma below, which follows from [11], Theorem 6.2, the statement (SS4) with ω = 0, and Corollary 6.6. Lemma 3.8.…”

Section: This Impliesmentioning

confidence: 99%

“…In paper [11], the definition of a Sheffer sequence on Φ was given and general properties of Sheffer sequences were studied. This research was motivated by numerous available examples of such sequences of polynomials: Hermite polynomials in Gaussian white noise analysis (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, we discuss the umbral and Sheffer operators. First, in Subsection 3.1, we recall the required definitions and results from [11], regarding Sheffer sequences on Φ . Next, in Subsection 3.2, we formulate and prove the main results of the paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

Finkelshtein

Kondratiev

Lytvynov

et al. 2019

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

For certain Sheffer sequences (s n ) ∞ n=0 on C, Grabiner (1988) proved that, for each α ∈ [0, 1], the corresponding Sheffer operator z n → s n (z) extends to a linear self-homeomorphism of E α min (C), the Fréchet topological space of entire functions of order at most α and minimal type (when the order is equal to α > 0). In particular, every function f ∈ E α min (C) admits a unique decomposition f (z) = ∞ n=0 c n s n (z), and the series converges in the topology of E α min (C). Within the context of a complex nuclear space Φ and its dual space Φ , in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on Φ . In particular, for Φ = Φ = C n with n ≥ 2, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space Φ , we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of E α min (Φ ) when α > 1. The latter result is new even in the one-dimensional case.Keywords: Infinite dimensional holomorphy; nuclear and co-nuclear spaces; sequence of polynomials of binomial type; Sheffer operator; Sheffer sequence; spaces of entire functions. 2010 MSC. Primary: 46E50, 46G20, 32A10, 32A15, 05A40. Secondary: 60H40.

show abstract

“…which is a formal series in powers of z ∈ C with coefficients from Φ. See [11] for further details on such formal power series.…”

Section: Holomorphic Mappings On Complex Nuclear and Co-nuclear Spacesmentioning

confidence: 99%

Section: Sheffer Sequences On φmentioning

confidence: 99%

Section: This Impliesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

Finkelshtein

Kondratiev

Lytvynov

et al. 2019

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

Stirling operators in spatial combinatorics

Finkelshtein

Kondratiev

Lytvynov

et al. 2022

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m) n can be extended from a natural number m ∈ N to the falling factorials (z) n = z(z−1) • • • (z−n+1) of an argument z from F = R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z) n through z k , k ≤ n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations-discrete Radon measures γ = i δ x i on X, where δ x i is the Dirac measure with mass at x i . The spatial falling factorials (γ, where M (n) (X) denotes the space of F-valued, symmetric (for n ≥ 2) Radon measures on X n . There is a natural duality between M (n) (X) and the space CF (n) (X) of F-valued, symmetric continuous functions on X n with compact support. The Stirling operators of the first and second kind, s(n, k) and S(n, k), are linear operators, acting between spaces CF (n) (X) and CF (k) (X) such that their dual operators, acting from M (k) (X) into M (n) (X), satisfy (ω) n = n k=1 s(n, k) * ω ⊗k and ω ⊗n = n k=1 S(n, k) * (ω) k , respectively. In the case where X has only a single point, the Stirling operators can be identified with Stirling numbers. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

show abstract

A biorthogonal approach to the infinite dimensional fractional Poisson measure

Bendong¹,

Menchavez²,

Silva³

2023

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

View full text Add to dashboard Cite

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure [Formula: see text], [Formula: see text], on the dual of Schwartz test function space [Formula: see text]. The Hilbert space [Formula: see text] of complex-valued functions is described in terms of a system of generalized Appell polynomials [Formula: see text] associated to the measure [Formula: see text]. The kernels [Formula: see text], [Formula: see text], of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system [Formula: see text], there is a generalized dual Appell system [Formula: see text] that is biorthogonal to [Formula: see text]. The test and generalized function spaces associated to the measure [Formula: see text] are completely characterized using an integral transform as entire functions.

show abstract

An infinite dimensional umbral calculus

Cited by 5 publications

References 34 publications

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

Stirling operators in spatial combinatorics

A biorthogonal approach to the infinite dimensional fractional Poisson measure

Contact Info

Product

Resources

About