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).     

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.     

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.     

A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Berührsatz). The same special Pascal theorem now leads to an incidence theorem, , for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.     

A transplantation theorem for Jacobi series in weighted Hardy spaces

A transplantation theorem for Jacobi series in weighted Hardy spaces is proved.     

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.     

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.     

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.     

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.     

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e ^h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \triangle_m{\triangle_{\mu}} has a positive lower bound λ₁(M) > 0 and the m(m > n)-dimensional Bakry-émery curvature is bounded from below by -\fracm-1m-2l₁(M){-\frac{m-1}{m-2}\lambda_1(M)}, then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.     

A large deviation theorem for weighted sums

Asymptotic representations are derived for large deviation probabilities of weighted sums of independent, identically distributed random variables. The main theorem generalizes a 1952 theorem of Chernoff which asserts that n ^–1 log P(S _n>cn)–log , where S _n is the partial sum of a sequence of independent, identically distributed random variables X ₁, X ₂, ... and is a constant depending on X ₁. The main result is similar in form to, but different in focus from, a particular case of Feller's (1969) theorem on large deviations for triangular arrays.This paper is based on work done for the author's doctoral dissertation written under Prof. Donald R. Truax of the University of Oregon, Eugene.     

A lower estimate for the Lebesgue constants of linear means of Laguerre expansions

S. G. Kal’neǐ derived in [5], [6] a quite sharp necessary condition for the multiplier norm of a finite sequence in the setting of Fourier-Jacobi series on L¹ with “natural weight” (which ensures a nice convolution structure). In this paper, Kalneǐ’s problem is considered in the setting of Laguerre series on weighted L¹-spaces; the admitted scale of weights contains in particular the appropriate “natural weights” occurring in transplantation and convolution.     

A heat kernel version of Cowling-Price theorem for the Laguerre hypergroup

In this paper, we prove a heat kernel version of Cowling-Price theorem for the Laguerre hypergroup.     

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,α_*), where K=[0,+∞[×R and α_* a convolution product on K coming from the product formula satisfied by the Laguerre functions (m∈N, α?0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μα_*k=μα_*?α_*μ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.     

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.     

A Tauberian theorem for weighted means in non-archimedean fields

In this note, K denotes a complete, non-trivially valued, non-archimedean field. We prove a Tauberian theorem for weighted means in K.     

Maximal function and multiplier theorem for weighted space on the unit sphere

For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex.     

A transplantation theorem for Fourier-Bessel coefficients

We prove a transplantation theorem for Fourier-Bessel coefficients. Theorems of such type were proved byAskey andWainger [1] andAskey [2] for ultraspherical and Jacobi coefficients, respectively. Our theorem can be also seen as a dual result to a transplantation theorem for Fourier-Bessel series which was proved byGilbert [3].     

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.