Hessenberg pairs of linear transformations

Let denote a field and V denote a nonzero finite-dimensional vector space over . We consider an ordered pair of linear transformations A:V→V and A^*:V→V that satisfy (i)–(iii) below.

1. [(i)]Each of A,A^* is diagonalizable on V.

2. [(ii)]There exists an ordering of the eigenspaces of A such that

where V_-1=0, V_d+1=0.

3. [(iii)]There exists an ordering of the eigenspaces of A^* such that

where , .

We call such a pair a Hessenberg pair on V. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.     

Über ein Turánsches problem für radiale, positiv definite Funktionen, II

Turán's problem is to determine the greatest possible value of the integral for positive definite functions f(x), , supported in a given convex centrally symmetric body , . We consider the problem for positive definite functions of the form f(x)=(x₁), , with supported in [0,π], extending results of our first paper from two to arbitrary dimensions.Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, ℓ-1 summability of the inverse Fourier integral on . Their investigations gave rise to a pair of transformations (h_d,m_d) on which they studied using special functions, in particular spherical Bessel functions.To study the d-dimensional Turán problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turán's problem.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

On partial sums of certain meromophic -valent functions

In this paper, we study the ratio of meromorphic p-valent functions in the punctured disk U^*={z:0<|z|<1} of the form to its sequence of partial sums of the form . Also, we determine sharp lower bounds for and .     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

The topological structure of fuzzy sets with endograph metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rⁿ, let be the family of all fuzzy sets ofRⁿ, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space with the topology of endograph metric is homeomorphic to the Hilbert cube Q=[-1,1]^ω iff Y is compact; and the space is homeomorphic to {(x_n)Q:sup|x_n|<1} iff Y is non-compact and locally compact.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

On wavelets related to the Walsh series

For any integers p,n≥2 necessary and sufficient conditions are given for scaling filters with pⁿ many terms to generate a p-multiresolution analysis in . A method for constructing orthogonal compactly supported p-wavelets on is described. Also, an adaptive p-wavelet approximation in is considered.     

Three solutions for a differential inclusion problem involving the -Laplacian

In this paper we consider differential inclusion problem involving the p(x)-Laplacian of the type

Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in .     

Solutions to operator equations on Hilbert -modules

Let be a C^*-algebra, E,F and G be Hilbert -modules, , and . We generalize the Douglas theorem about the operator equation TX=T^′ from Hilbert space to Hilbert C^*-module. To the equation TX=T^′ and to the system of two equations TX=T^′ and XS=S^′, we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G=E). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G=E) to the equation TX=T^′ are given too.     

Asymptotic structure and the existence of noncompact operators between Banach spaces

We investigate the problem of the existence of a noncompact operator T:X₀X→Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for and , let T_ξ,η be the linear map which sends each x_i to y_i. We prove that if for some then every T:X₀X→Y is compact. If for n=2 all such maps have norm 1 we show the existence of a noncompact T:X₀X→Y.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.