A generalized Chebyshev theory

In this paper we develop, without assuming the Haar condition, a generalized Chebyshev theory for Chebyshev approximation which is similar to the classical Chebyshev theory and contains it as a special case. The Project Supported by National Natural Science Foundation of China     

Tensor Trains Approximation Estimates in the Chebyshev Norm

Computational Mathematics and Mathematical Physics - A new elementwise bound on the cross approximation error used for approximating multi-index arrays (tensors) in the format of a tensor train is...     

Fractional Moments of Automorphic L-Functions on GL（m）

Let π be an irreducible unitary cuspidal representation of GLm(AQ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator B _n fixing a strictly positive function f ₀, and a second function f ₁ such that f ₁/f ₀ is strictly increasing, within the framework of extended Chebyshev spaces U _n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B _n : C[a, b] → U _n with strictly increasing nodes, fixing f₀, f₁ ? U_n{f_{0}, f_{1} \in U_{n}} . If U_n ì U_{n + 1}{U_{n} \subset U_{n + 1}} and U _n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B _n+1 : C[a, b] → U _n+1 with strictly increasing nodes, fixing f ₀ and f ₁. In particular, if f ₀, f ₁, . . . , f _n is a basis of U _n such that the linear span of f ₀, . . . , f _k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B _n with increasing nodes fixing f ₀ and f ₁. The second main result says that under the above assumptions the following inequalities hold

Duality Gap Estimates for Weak Chebyshev Greedy Algorithms in Banach Spaces

Computational Mathematics and Mathematical Physics - The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept...     

Chebyshev lattices,a unifying framework for cubature with Chebyshev weight function

We present a multivariate extension to Clenshaw-Curtis quadrature based on Sloan’s hyperinterpolation theory. At the centre of it, a cubature rule for integrals with Chebyshev weight function is needed. We introduce so called Chebyshev lattices as a generalising framework for the multitude of point sets that have been discussed in this context. This framework provides a uniform notation that extends easily to higher dimensions. In this paper we describe many known point sets as Chebyshev lattices.     

N-symmetric Chebyshev polynomials in a composite model of a generalized oscillator

We continue to study a composite model of a generalized oscillator generated by an N-periodic Jacobi matrix. The foundation of the model is a system of orthogonal polynomials connected to this matrix for N = 3, 4, 5. We show that such polynomials do not exist for N ?? 6.     

Estimates for generalized derivatives of quasipolynomials from Muntz system

L. Schwartz has obtained estimates for coefficients of quasipolynomials from Muntz system. In this paper we obtain exact estimates for generalized derivatives of such quasipolynomials on the interval [0, 1].     

A formula for the Chebyshev PSI function

A formula expressing the Chebyshev function in terms of the characteristic values of the Laplace-Beltrami operator on the fundamental domain of a modular group and the hyperbolic classes of conjugate elements of this group is derived.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 497–503, April, 1978.     

On the generalized Chebyshev polynomials corresponding to plane trees of diameter 4

The combinatorial classification of plane trees by the number of realizations of their valency sets has distinguished some special classes of plane trees. One of them, the plane trees of diameter 4, turned out to be a very interesting object of investigation from the Galois action point of view. In this paper, we present equation sets for some subclasses of trees of diameter 4, calculate discriminants of the corresponding generalized Chebyshev polynomials, some related polynomials, and their fields of definitions, and use this to get some information about the Galois action on plane trees. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 19–33, 2007.     

Construction of Graded Covariant GL(m/n) Modules Using Tableaux

Irreducible covariant tensor modules for the Lie supergroups GL(m/n) and the Lie superalgebras gl(m/n) and sl(m/n) are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on V ^l . The isomorphism between this group of actions and the symmetric group S _l enables the graded generalization of the Young symmetrizers, and hence of the column relations and Garnir relations, to be made. Consequently, corresponding to each partition of l an irreducible GL(m/n) module may be obtained as a submodule of V ^l . A basis for the module labeled by the partition is provided by GL(m/n)–standard tableaux of shape defined by Berele and Regev. The reduction of an arbitrary tableau to standard form is accomplished through the use of graded column relations and graded Garnir relations. The standardization procedure is algorithmic and allows matrix representations of the Lie superalgebras gl(m/n) and sl(m/n) to be constructed explicitly over the field of rational numbers. All the various steps of the standardization algorithm are exemplified, as well as the explicit construction of matrices representing particular elements of gl(m/n) and sl(m/n).     

Remarks on a generalized beta function

We show that a certain generalized beta function B(x,y;b) which reduces to Euler's beta functions B(x,y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a finite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B(x,y;b) may be represented as an infinite series of either Wittaker functions or Laguerre polynomials. As a byproduct of this investigation we deduce representations for several infinite series containing Wittaker functions, Laguerre polynomials, and products of both.     

一般线性李超代数及其超群的无穷小子群的表示

在索特征代数闭域上考虑一般线性李超代数gl(m |n)的限制表示与超群GL(m | n)的有理表示以及它们的关系.主要结果为: (1)对gl(m | n)的不可约限制表示进行分类,其中某些单模恰是Kac-模.类似于复数域情形,给出了Kac-模不可约的充要条件; (2)当m≠n(mod p)以及p≥2h-2(h=max{m,n})时,gl(m | n)的限制投射模可以被提升为有理GL(m | n)-模,并且证明了不可约表示的投射覆盖具有Z-滤过,即滤过中的每个子商同构于"baby Vlerma模";(3)得到了一般线性超群G=GL(m | n)的r阶nobenius核的反转公式,它反映了单Gγ-模的投射覆盖的Z-滤过重数与广义baby Verma模的合成因子效之间的关系.