Schurian vector space categories, hereditary algebras and Roiter's norm

We associate a positive real number to any vector space K-category over a field K. Generalizing a result of Nazarova and Roiter, we show that a schurian vector space K-category is representation-finite if and only if is finite and . Such vector space categories are quasilinear, i.e. its indecomposables are simple modules over their endomorphism ring. Recently, Nazarova and Roiter introduced the concept of -faithful poset in order to clarify the structure of critical posets. Their conjecture on the precise form of -faithful posets was established by Zeldich. We generalize these results and characterize -faithful quasilinear vector space K-categories in terms of a class of hereditary algebras H_ρ(D) parametrized by a skew-field D and a rational number ρ1.     

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.     

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.     

Approximation theorems for localized szsz–Mirakjan operators

In the present paper, we investigate the convergence and the approximation order of the localized Szsz–Mirakjan operators, and obtain some new results to improve the results due to Omey [Note on operators of Szsz–Mirakjan type, J. Approx. Theory 47 (1986) 246–254].     

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.     

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.     

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.     

A non-splitting theorem in the enumeration degrees

We complete a study of the splitting/non-splitting properties of the enumeration degrees below by proving an analog of Harrington’s non-splitting theorem for the enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.     

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.     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

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 .     

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.     

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.     

Pseudoprime factorizations of integer matrices

We give a method of factoring integer matrices in into components such that the factorization is not unique unless certain information is known. In Section 2, we introduce this method of factorization and provide theorems which establish its well-definedness. In Section 3, we construct a matrix in as a product of specific types of matrices and establish an algorithm for factoring the result uniquely given an amount of information.     

Linear information versus function evaluations for -approximation

We study algorithms for the approximation of functions, the error is measured in an L₂ norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n^-b.Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether p>b is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Woźniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that implies . Here we prove that the case and b=0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case is quite different, it is still open whether b=p always holds in that case.     

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.

Representation type of the blocks of category in types and

In this paper a complete classification of the representation type of the infinitesimal blocks of the parabolic category for the complex simple Lie algebras of types F₄ and G₂ is given.     

Shape preserving approximation on the real line with exponential weights

The well-known “Bernstein’s weighted problem” deals with the possibility of weighted approximation on the whole real line. In this paper, we show the possibility of k-monotone approximation on the real line with Freud’s weight .     

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.