On optimal kernel choice for deconvolution

In this note we show that, from a conventional viewpoint, there are particularly close parallels between optimal-kernel-choice problems in non-parametric deconvolution, and their better-understood counterparts in density estimation and regression. However, other aspects of these problems are distinctly different, and this property leads us to conclude that “optimal” kernels do not give satisfactory performance when applied to deconvolution. This unexpected result stems from the fact that standard side conditions, which are used to ensure that the familiar kernel-choice problem has a unique solution, do not have statistically beneficial implications for deconvolution estimators. In consequence, certain “sub-optimal” kernels produce estimators that enjoy both greater efficiency and greater visual smoothness.     

Concentration Problems for Bandpass Filters in Communication Theory over Disjoint Frequency Intervals and Numerical Solutions

The concentration problem of maximizing signal strength of bandlimited and timelimited nature is important in communication theory. In this paper we consider two types of concentration problems for the signals which are bandlimited in disjoint frequency-intervals, which constitute a band-pass filter. For the first type the problem is to determine which members of L ²(−∞,∞) lose the smallest fraction of their energy when first timelimited and then bandlimited. For the second type the problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted to a given time interval. For both types of problems, basic theoretical properties and numerical algorithms for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L ²(−∞,∞) are also proved. Numerical computations are carried out which corroborate the theory. Relationship between eigenvalues of these two types of problems is also established. Several properties of eigenvalues of both types of problems are proved.     

Half Sampling on Bipartite Graphs

On a bipartite graph G we consider the half sampling problem of uniquely recovering a function from its values on the even vertices, under the appropriate half bandlimited assumption with respect to a Laplacian on the graph. We discuss both finite and infinite graphs, give the appropriate definition of “half bandlimited” that involves splitting the mid frequency, and give an explicit solution to the problem. We discuss in detail the example of a regular tree. We also consider a related sampling problem on graphs that are generated by edge substitution.     

On the determination of the worst sampling error in a communication system for pulse-amplitude-modulated signals by solving an approximation problem

A communication system for pulse-amplitude-modulated signals is considered whose impulse response is bandlimited. For a certain class of sampling errors the worst case is investigated where the sampling error of the system is such that the least mean square transmission error achieved by a suitable choice of the impulse response becomes maximal. This problem is dualized in the sense of two person game theory and thereby transferred to an equivalent approximation problem for whose solution a necessary and sufficient condition is given and applied to special cases.     

Deconvolution and Regularization with Toeplitz Matrices

By deconvolution we mean the solution of a linear first-kind integral equation with a convolution-type kernel, i.e., a kernel that depends only on the difference between the two independent variables. Deconvolution problems are special cases of linear first-kind Fredholm integral equations, whose treatment requires the use of regularization methods. The corresponding computational problem takes the form of structured matrix problem with a Toeplitz or block Toeplitz coefficient matrix. The aim of this paper is to present a tutorial survey of numerical algorithms for the practical treatment of these discretized deconvolution problems, with emphasis on methods that take the special structure of the matrix into account. Wherever possible, analogies to classical DFT-based deconvolution problems are drawn. Among other things, we present direct methods for regularization with Toeplitz matrices, and we show how Toeplitz matrix–vector products are computed by means of FFT, being useful in iterative methods. We also introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate.     

Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.     

Factorization and imbedding for general linear boundary value problems

A new direction in the theory of general linear boundary value problems is explored. The starting point is an explicit Volterra factorization of the Green's matrix (and related kernels) associated with the problem. This result leads to (1) imbedding of the boundary value problems, (2) initial value algorithms for their solution, and (3) comparison theorems relating two different boundary value problems with a common boundary condition. Extensions and connections with earlier work in this area are presented.     

On identification of memory kernels in linear theory of heat conduction

A linear integrodifferential equation describing the heat flow in a material with memory is considered. This equation contains a pair of time-dependent convolution kernels that are unknown. Such kernels are determined as solutions of an optimal control problem by using additional data obtained from measurements of average temperature around some fixed points of the domain over some finite time interval. We show the existence of an optimal solution of this problem and derive optimality conditions for it.     

Triple trigonometric integral equations with an application

Closed form solution has been obtained for certain triple integral equations with trigonometric kernels which arise in the crack and punch problems in the linear theory of elasticity. As an application solution is obtained for a two dimensional punch problem.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

A Multisensor Deconvolution Problem

Meisters and Peterson gave an equivalent condition under which the multisensor deconvolution problem has a solution when there are two convolvers, each the characteristic function of an interval. In this article we find additional conditions under which the deconvolution problem for multiple characteristic functions is solvable. We extend the result to the space of Gevrey distributions and prove that every linear operator S, fromthe space of Gevrey functions with compact support onto itself, which commutes with translations can be represented as convolution with a unique Gevrey distribution T of compact support. Finally, we find explicit formula for deconvolvers when the convolvers satisfy weaker conditions than the equivalence conditions using nonperiodic sampling method.     

The optimal solution of multi-kernel regularization learning

In regularized kernel methods, the solution of a learning problem is found by minimizing a functional consisting of a empirical risk and a regularization term. In this paper, we study the existence of optimal solution of multi-kernel regularization learning. First, we ameliorate a previous conclusion about this problem given by Micchelli and Pontil, and prove that the optimal solution exists whenever the kernel set is a compact set. Second, we consider this problem for Gaussian kernels with variance σ∈(0,∞), and give some conditions under which the optimal solution exists.     

Algorithm for min-range multiplication of affine forms

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.     

A system of boundary integral equations for orthotropic shells of positive curvature with slits and holes

We propose a method of constructing a system of boundary integral equations for the problem of the stress state of an orthotropic shell with slits and holes. Using the theory of distributions and the two-dimensional Fourier transform, we reduce the problem to a system of boundary integral equations. In the solution obtained the kernels of the system of integral equations do not contain the direction cosines of the unit outward normal vector explicitly. There are no extra-integral terms. The matrix of the kernels is symmetric. The kernels are regular or have a logarithmic singularity. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 59–69.     

Data Pulse Design--Intersymbol Interference Aspects

Past theoretical intersymbol interference studies aimed at findingoptimal overall network characteristics have not given muchattention to design limitations. In this paper we address theproblem of identifying, from an ensemble of filters, that optimalfilter design which maximizes a sampling signal to intersymbolinterference noise power ratio. Sufficient conditions are givenfor such non-linear extremal problems to be well posed. Continuingunder mild assumptions, attention is then focused on networksof parallel structure. In this case a procedure for constructinga solution is given. The parallel structure solution providesa partial solution to more complex network design problems.Examples of R–C and bandlimited network ensembles areconsidered in detail. Finally it is shown that these non-linearextremal problems generally lie outside the realm of more standardfrequency domain norm minimization problems such as weightedleast squares and minimax. The development is general enough to encompass design parametershaving components in function space.     

The Multichannel Deconvolution Problem: A Discrete Analysis

We study the multichannel deconvolution problem (MDP) in a discrete setting by developing the theory for converting the method used in the continuous setting in [36]. We give a method for solving the MDP when the convolvers are characteristic functions, derive the explicit form of the linear system, and obtain an upper bound on the condition number of the system in a particular case. We compare the Schiske reconstruction [28] to our solution in the discrete setting, and give an explicit formula for the corresponding error. We then give the algorithm for solving the general MDP and discuss in detail the local reconstruction aspects of the problem. Finally, we describe a method for improving the reconstruction by regularization and give some explicit estimates on error bounds in the presence of noise.     

An axisymmetric problem of viscoelasticity

The problem of a long hollow viscoelastic cylinder, enclosed in an elastic shell and loaded by internal pressure, is considered. Two cases are investigated: 1) the shear and volume relaxation kernels are proportional; 2) the shear and volume creep kernels are proportional. Resolving integral equations with singular kernels of a nondifference type are obtained and a method of solution is proposed. The convergence of the method is proved.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 5, pp. 798–805, September–October, 1969.     

Prolate spheroidal wave functions on a disc—Integration and approximation of two-dimensional bandlimited functions

We consider the problem of integrating and approximating 2D bandlimited functions restricted to a disc by using 2D prolate spheroidal wave functions (PSWFs). We derive a numerical scheme for the evaluation of the 2D PSWFs on a disc, which is the basis for the numerical implementation of the presented quadrature and approximation schemes. Next, we derive a quadrature formula for bandlimited functions restricted to a disc and give a bound on the integration error. We apply this quadrature to derive an approximation scheme for such functions. We prove a bound on the approximation error and present numerical results that demonstrate the effectiveness of the quadrature and approximation schemes.