Hypercircle inequalities and sampling theory

In this article the well-known hypercircle inequality is extended to the Riesz bases setting. A natural application for this new inequality is given by the estimation of the truncation error in nonorthogonal sampling formulas. Examples including the estimation of the truncation error for wavelet sampling expansions or for nonorthogonal sampling formulas in Paley–Wiener spaces are exhibited.     

A General Framework of Multivariate Wavelets with Duals

A comprehensive development of multivariate wavelets along with their duals is presented in this paper. The basic ingredients, such as duality relations, reconstruction and decomposition formulas, and the notion of infinite direct sums, are formulated and established in the general nonorthogonal and multivariate setting. Special emphases include duality criteria and stability conditions. As an application, new results are contributed, particularly in dual wavelets, to the existing literature on low-dimensional wavelets. In addition, the special case when the dilation matrix has determinant 2 is studied in some detail.     

An Estimation of the Truncation Error for the Two-Channel Sampling Formulas

The aim of the article is to obtain an estimation for the truncation error in the two-channel sampling formulas. Since these formulas are expansions with respect to suitable Riesz bases in Paley-Wiener spaces, the truncation error will be estimated by using the hypercircle inequality in the Riesz bases setting. In so doing, the norm of an involved operator is calculated, and the remainder of the series of the absolute square sampling functions is estimated.     

Characterizing consistency in probabilistic logic for a class of Horn clauses

The problem of consistently assigning probabilities to logical formulas is an important problem. In this paper a set of logical formulas will be identified for which the problem can be solved. For every directed graph we define a set of logical formulas that it represents. If the underlying (undirected) graph is either perfect or t-perfect a closed form solution to the consistency problem can be given. A remarkable property of the class of formulas identified here is that it turns out to be closed under duality (if a set of formulas is represented by a digraph then the dual set of formulas is also represented by a digraph).     

On Lagrange-Type Interpolation Series and Analytic Kramer Kernels

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. A challenging problem is to characterize the situations when these sampling formulas can be written as Lagrange-type interpolation series. This article gives a necessary and sufficient condition to ensure that when the sampling formula is associated with an analytic Kramer kernel, then it can be expressed as a quasi Lagrange-type interpolation series; this latter form is a minor but significant modification of a Lagrange-type interpolation series. Finally, a link with the theory of de Branges spaces is established. Received: October 8, 2007. Revised: December 13, 2007.     

Calculation formulas of linear geometrical spreading for ray tracing in a 3D block inhomogeneous gradient medium

Suitable recurrent formulas are derived for direct programming of linear geometrical spreading of a central field of seismic rays in a 3D block gradient medium. These formulas are needed to organize shooting for area observation systems. For recalculation formulas across an interface, a new representation which involves a special nonorthogonal projection operator is found. This operator allows for additive separation of terms that depend only on the ray curvature, boundary curvature, and variable character of the velocity ratio along the boundary. Formulas which express partial derivatives of the eikonal via linear and angular geometrical spreadings are presented.     

Local analysis,cardinality, and split trick of quasi-biorthogonal frame wavelets

The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.     

Testing properties of graphs and functions

We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. We give a characterization of properties testable this way, and extend a number of results about “large graphs” to this setting.     

A fresh approach to the Paley–Wiener theorem for Mellin transforms and the Mellin–Hardy spaces

Here we give a new approach to the Paley–Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar‐analytic function in the Mellin setting and Mellin–Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.     

Multivariate generalized sampling in shift-invariant spaces and its approximation properties

Nowadays the topic of sampling in a shift-invariant space is having a significant impact: it avoids most of the problems associated with classical Shannon's theory. Under appropriate hypotheses, any multivariate function in a shift-invariant space can be recovered from its samples at Zd. However, in many common situations the available data are samples of some convolution operators acting on the function itself: this leads to the problem of multivariate generalized sampling in shift-invariant spaces. This extra information on the functions in the shift-invariant space will allow to sample in an appropriate sub-lattice of Zd. In this paper an L²(Rd) theory involving the frame theory is exhibited. Sampling formulas which are frame expansions for the shift-invariant space are obtained. In the case of overcomplete frame formulas, the search of reconstruction functions with prescribed good properties is allowed. Finally, approximation schemes using these generalized sampling formulas are included.     

Projections for harmonic Bergman spaces and applications

On the setting of general bounded smooth domains in , we construct L¹-bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic Bergman spaces. In addition, we show that those projections satisfy Sobolev L^p-estimates of any order even for p=1. Among applications are Gleason's problems for the harmonic Bergman-Sobolev and (little) Bloch functions on star-shaped domains with strong reference points.     

A new approach to the theory of classical hypergeometric polynomials

In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.

     

Sampling multipliers and the Poisson Summation Formula

Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined sense. Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces. These Poisson Summation Formulas can be used to prove corresponding sampling theorems. The sampling operators used to understand and prove the aforementioned Poisson Summation Formulas lead to the introduction of spaces of continuous linear operators which commute with integer translations. Operators L of this type are appropriately called sampling multipliers. For a given function f, they give rise to new sampling formulas, whose sampling coefficients are of the form Lf. In practice, Lf can be used to model noisy data or data where point values are not available. By representation theorems of the second named author, some of these operator spaces are proved to be mixed norm spaces. The approach and results of this paper were developed in the context of Duffin and Schaeffer’s theory of frames. In particular, sampling multipliers L are related to the Bessel map used by Duffin and Schaeffer in their definition of the frame operator. The first named author was supported in part by AFOSR contract F49620-96-1-0193. The second named author was supported by the Cusanuswerk.     

Regularity properties of semigroups generated by some Fleming-Viot type operators

We study different qualitative properties of the semigroup generated by some degenerate differential elliptic operators on the standard simplex of Rd. Some methods are new and are based on the representation formulas of the semigroup in terms of iterates of suitable positive operators. The main result is the ultracontractivity property which is obtained in the setting of weighted Lp-spaces. We describe the asymptotic behavior of the semigroup and obtain the compactness property in the same setting and also in spaces of continuous functions.     

On progressive functions

Progressive functions at time t involve only the progressive functions at time before t and some nice compactly supported function at time t. We give sufficient conditions and explicit formulas to construct progressive functions with exponential decay and characterize the conditions on which the positive integer translates of a progressive function are orthonormal or a Riesz sequence. We provide explicit ways for construction of orthonormal progressive functions and for construction of the biorthogonal functions of nonorthogonal progressive functions. Such progressive functions can be used to construct wavelets with arbitrary smoothness on the half line if they are generated by a smooth refinable compactly supported function.     

Separable Systems of Coordinates for Triangular Newton Equations

Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic with respect to velocities integral of motion, it usally implies existence of further n − 1 integrals that are also quadratic. These integrals make the triangular system separable in new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and noconfocal and can intersect along lower dimensional singular manifolds. We present here separability theory for n -dimensional triangular systems and analyze the structure of separation coordinates in two and three dimensions.     

Solution of the heat equation on unstructured curvilinear grids

A computational approach to the solution of the heat equation is proposed. In the case of three-dimensional oblique (nonorthogonal) unstructured grids, this approach results in a compact grid stencil and unconditionally stable computational algorithm. A feature of the proposed approach is the use of flux functions as dependent separate variables. Mainly hexagonal grids are considered in which every cell can be continuously mapped onto a unit cube. Computational examples are presented.     

四元数体上若干线性代数问题的显式

通过建立四元数乘积的一个弱可交换律,分别给出四元数体上的线性方程组的解和克莱姆解式、向量的相关性、矩阵的逆与秩以及线性变换的特征根与特征向量等存在性的充要条件,从而得到这些问题的一种有效计算方法．     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.