Generalized Virasoro Operators

Abstract

We define generalized Virasoro operators acting on a Fock space V(Γ). These generalize the standard construction of Virasoro operators. By using the Jacobi Identity we compute the commutators of these operators. These operators result in an abelian extension of the toroidal Lie algebra. We explicitly describe the abelian extension.     

Spectral Gaps Resulting from Periodic Pertubations of a Class of Jacobi Operators

We investigate the spectral properties of a class of Jacobi matrices resulting from periodic pertubations of Jacobi operators with smooth coefficients.     

SINGULAR INTEGRAL OPERATORS IN L~2 SPACES ASSOCIATED WITH THE GENERALIZED JACOBI WEIGHTS

In this paper, a kind of new definitions of singular integral operators in the weighted L2 space with the generalized Jacobi weights are given, then the boundedness in the weighted L2 space, the relationships between these operators and the classic singular integral operators are proved. For the convenience in the later use, similar results in the L2 space with Jacobi weights are given.     

Smoothness of density of states for random decaying interaction

In this paper we consider one dimensional random Jacobi operators with decaying independent randomness and show that under some condition on the decay vis-a-vis the distribution of randomness, that the distribution function of the average spectral measures of the associated operators are smooth.     

A Generalization of Hörmander's Inequality-I

We give sufficient conditions to generalize Hörmander's inequality to the case of operators with multiple characteristics of order higher than two     

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.     

Polynomial operators and local smoothness classes on the unit interval

We obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded.     

某些多元线性正算子的加权逼近

本文首先给出了在Ｌｐ逼近意义下某些线性正算子加Ｊａｃｏｂｉ权逼近时的特征定理，作为应用，我们给出了多元Ｂａｓｋａｋｏｖ型算子、多元Ｓｚａｓｚ－Ｍｉｒａｋｊａｎ型算子和多元Ｂｅｔａ算子加权逼近时的特征刻划．     

关于一类Hermite-Fejér插值算子的平均收敛

本文给出基于{xk}_(k=0)~(n+1)的Hermite-Fejér插值算子平均收敛的一些新结论,这里x0=1,xn+1=-1,xk（k=1,2,…,n）是n阶Jacobi多项式的零点.     

Bernstein型算子加Jacobi权逼近

对于Bernstein型算子,证明它在通常的加权范数下是无界的,通过引进新的加权范数,研究其加Jacobi权的逼近性质,得到加权逼近的正逆定理,从而导出加权逼近特征的等价刻画．     

Bernstein型算子线性组合加Jacobi权逼近及高阶导数的等价定理

本文利用加权Ditzian-Totik光滑模证明Bernstein型算子的线性组合加权逼近阶估计和等价定理;同时,研究加Jacobi权下Benstein型算子的高阶导数与所逼近函数光滑性之间的关系.     

关于多元Szász-Mirakjan算子的加权逼近

本文主要讨论一类二元Ｓｚáｓｚ－Ｍｉｒａｋｊｉａｎ算子的加权逼近问题。我们首先指出了在通常的加权范数下它是无界的。然后我们给出了一类新的加权范数，在此范数下，它是有界的。最后利用一类多元Ｋ－泛函与多元分解技巧，我们给出了二元Ｓｚａｓｚ－Ｍｉｒａｋｉａｎ算子加权逼近的特征刻划。关键词     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.     

Essential Closures and AC Spectra for Reflectionless CMV,Jacobi, and Schrödinger Operators Revisited

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures.As an elementary application of the notion of the essential closure of subsets of ? we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set ? of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure \({\overline{\mathcal{E}}}^{e}\) of the set ? (with uniform multiplicity two on ?). Though this result in the case of Schrödinger and Jacobi operators is known to experts, we feel it nicely illustrates the concept and usefulness of essential closures in the spectral theory of classes of reflectionless differential and difference operators.     

A commutator approach to absolute continuity for unbounded Jacobi operators

This paper uses commutator equations to study the absolute continuity of spectral measures associated with certain subclasses of unbounded self-adjoint Jacobi matrix operators determined by properties of the diagonal and subdiagonal sequences. If the diagonal sequence is the zero sequence, properties of the difference sequence of the subdiagonal determine the choice of a bounded operator for the commutator equation. The structure of the resulting commutator leads to results on absolute continuity.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted L ^p spaces. This paper is partially supported by MTM2004/05878. Third and fourth authors are also partially supported by grant PI042004/067.     

New approach to differential equations with countable impulses

This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular flows on Riemannian manifolds with dimension n + 1.A new class of generalized differential operators is defined.We investigate the kernel of the corresponding maximal operators by applying operator theory.It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic,in which there are n dimension solutions with exponential...