Multipliers of Laplace transform type for ultraspherical expansions

We define and investigate the multipliers of Laplace transform type associated to the differential operator L_λf (θ) = –f ″(θ) – 2λ cot θf ′(θ) + λ²f (θ), λ > 0. We prove that these operators are bounded in L^p ((0, π), dm_λ) and of weak type (1, 1) with respect to the same measure space, dm_λ (θ) = (sin θ)^2λ dθ. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A localization theorem for Laguerre expansions

Regularity properties of Laguerre means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localization theorem for Laguerre expansions.     

On Riesz transforms for Laguerre expansions

We prove that Riesz transforms and conjugate Poisson integrals associated with the multi-dimensional Laguerre semigroup are bounded in Lp,1<p<∞. Our main tools are appropriately defined square functions and the Littlewood-Paley-Stein theory.     

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

Generalized functions and Laguerre expansions

The number of spanning trees in a class of directed circulant graphs with generators depending linearly on the number of vertices \(\beta n\), and in the nth and \((n-1)\)th power graphs of the \(\beta n\)-cycle are evaluated as a product of \(\lceil \beta /2\rceil -1\) terms.     

Summability of Laguerre expansions

В РАБОтЕ пОлУЧЕНы УсИ лЕНИь РЕжУльтАтОВ МА РкЕттА О сУММИРУЕМОстИ МОДИФ ИцИРОВАННых РАжлОжЕНИИ лАгЕРРА. У стАНОВлЕНО, ЧтОα=1/6 Ест ь кРИтИЧЕскИИ ИНДЕкс Д ль сУММИРУЕМОстИ пО ЧЕжАРО. ДОкАжАНО, Чт О пРИα=1/6 сРЕДНИЕ ЧЕжАР О схОДьтсь пОЧтИ ВсУДУ. пОлУЧЕН тАкжЕ АНАлОг тЕОРЕМы ФЕИЕРА-лЕБЕгА.     

Riesz transforms for multi-dimensional Laguerre function expansions

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, the appropriately defined Riesz-Laguerre transforms , j=1,2,…,d, are Calderón-Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz-Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy-Riemann type and to recover the Riesz-Laguerre transforms on the boundary.     

Laplace Type Multipliers for Laguerre Expansions of Hermite Type

We investigate Laplace transform type and Laplace-Stieltjes type multipliers associated to the multi–dimensional Laguerre function expansions of Hermite type. We prove that, under the assumption α _i ≥ ?1/2, α _i ? (?1/2, 1/2), these operators are Calderón-Zygmund operators. Consequently, their mapping properties follow by the general theory.     

A weighted transplantation theorem for Laguerre function expansions

We establish a generalized weighted transplantation theorem for Laguerre function expansions, which extends the corresponding result by G. Garrigós et al. “A sharp weighted transplantation theorem for Laguerre function expansions” (J. Funct. Anal. 244 (2007), pp. 247–276).     

On Connections Between Hankel, Laguerre and Heisenberg Multipliers

We prove two results showing connections between Hankel multipliers,multipliers for a certain kind of Laguerre expansion and multipliersfor the Heisenberg sublaplacian. In their spirit they are closeto the well-known theorems of de Leeuw and Igari relating L^pmultipliers on R and its discrete subgroup Z. Also a conjectureis stated and its consequences are discussed.     

A sharp weighted transplantation theorem for Laguerre function expansions

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in Lp spaces with power weights. Namely, the operators interchanging and are bounded in Lp(yδp) if and only if , where ρ=min{α,β}. This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ?0. Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in Lp(yδp).     

The bilateral Laguerre transform

In the earlier paper [13], the crucial role of the complex plane in the formulation of the algorithms was evident, even though the algorithms were entirely in the real domain. For the bilateral transform, the complex plane is again very much present, with Laurent expansions, bilateral Laplace transformation, and conformal mapping entering as crucial tools. The first section extends the earlier formalism to the full continuum. That this extension is natural, and not just an artificial piecing together of the formalism for each half line, will be clear from (1.9), (1.12), and (1.13). The harmony of the basis will also emerge vividly in Sec. 3, which deals with the extent of the transform coefficients and associated uncertainty relations. The topic of extent is crucial to the utility of the Laguerre-transform method as a numerical tool. Numerical examples are presented in Sec. 5.     

Triebel-Lizorkin spaces associated with Laguerre and Hermite expansions

It is proved that Triebel-Lizorkin spaces for some Laguerre and Hermite expansions are well-defined.

     

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, αi∉(−1/2,1/2), the appropriately defined Riesz transforms , j=1,2,…,d, are Calderón-Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α∈(−1/2,1/2), by means of a local Calderón-Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy-Riemann type equations and to recover the Riesz-Laguerre transforms on the boundary. The two specific values of α, (−1/2,…,−1/2) and (1/2,…,1/2), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.     

Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials \(L_{n}^{(\alpha )}(x)\), as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of \(L_{n}^{(\alpha )}(x)\) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided.     

Weyl fractional calculus and Laplace transform

The Weyl fractional calculus is developed to obtain Laplace transforms oft ^q ?(t) (for all real values ofq) where ?(t) is taken in the form off(a√(t ²?b ²)) and certain other forms. Also, a generating function involvingH-function of several variables is established with the help of generalized Taylor series.     

Laguerre polynomial expansions in indefinite inner product spaces

When ?j ? 1 < α < ?j, where j is a positive integer, the Laguerre polynomials {L_n^(α)}_{n = 0}^∞ form a complete orthogonal set in a nondegenerate inner product space H which is defined by employing an appropriate regularized linear functional on H^(j)[[0, ∞); x^{α + j}e^?x]. Expansions in terms of these Laguerre polynomials are exhibited. The Laguerre differential operator is shown to be self-adjoint with real, discrete, integer eigenvalues. Its spectral resolution and resolvent are exhibited and discussed.     

Hardy spaces and Laguerre expansions on the dual of the Heisenberg group

Sia D il dominio di Siegel e H_n il gruppo di Heisenberg. Si considera il sistema ortogonale ottenuto dai monomi normalizzati delta palla unitaria di ⁿ⁺¹ tramite la trasformata di Cayley. La trasformata di Fourier di tali funzioni ristrette ad H_n viene esplicitamente calcolata. Rispetto alla base di Hermite di L²( ⁿ ), si ottengono operatori di rango uno espressi in termini di funzioni di Laguerre.