Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

NON-STANDARD COMMUTATORS FOR ROUGH OSCILLATORY SINGULAR INTEGRALS

In this paper, the author studies a class of non-standard commutators with higher order remainders for oscillatory singular integral operators with phases more general than polynomials. For 1 〈 p 〈 ∞, the L^p-boundedness of such operators are obtained provided that their kernels belong to the spaces L^q（s^n-1） for some q 〉 1.     

Schur Q-polynomials, multiple hypergeometric series and enumeration of marked shifted tableaux

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.     

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}_t>0, are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron-Martin type quasi-invariance results, (3) good Lp-bounds on the corresponding Radon-Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.     

A two-dimensional matrix Padé-type approximation in the inner product space

By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant (BMPTA) in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of BMPTA is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of BMPTA is applied to partial realization problems of 2-D linear systems.     

An extension of Warnaar's matrix inversion

We present a necessary and sufficient condition for two matrices given by two bivariate functions to be inverse to each other with certainty in the cases of Krattenthaler formula and Warnaar's elliptic matrix inversion. Immediate consequences of our result are some known functions and a constructive approach to derive new matrix inversions from known ones.

     

Shannon sampling II: Connections to learning theory

We continue our study [S. Smale, D.X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. 41 (2004) 279–305] of Shannon sampling and function reconstruction. In this paper, the error analysis is improved. Then we show how our approach can be applied to learning theory: a functional analysis framework is presented; dimension independent probability estimates are given not only for the error in the L² spaces, but also for the error in the reproducing kernel Hilbert space where the learning algorithm is performed. Covering number arguments are replaced by estimates of integral operators.     

Volume growth and heat kernel estimates for the continuum random tree

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.      

Holomorphic Lefschetz Fixed Point Formula for Non-compact K¨ahler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” K¨ahler manifolds (e.g. K¨ahler hyperbolic manifolds, bounded domains of holomorphy) by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.