A new approach to the convolution operator on a finite interval

The convolution operator on a finite interval defined on a space ofL ₂ functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane . The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.     

Asymptotic bias and variance for a general class of varying bandwidth density estimators

Summary We consider a general class of varying bandwidth estimators of a probability density function. The class includes the Abramson estimator, transformation kernel density estimator (TKDE), Jones transformation kernel density estimator (JTKDE), nearest neighbour type estimator (NN), Jones-Linton-Nielsen estimator (JLN), Taylor series approximations of TKDE (TTKDE) and Simpson's formula approximations of TKDE (STKDE). Each of these estimators needs a pilot estimator. Starting with an ordinary kernel estimator , it is possible to iterate and compute a sequence of estimates , using each estimate as a pilot estimator in the next step. The first main result is a formula for the bias order. If the bandwidths used in different steps have a common orderh=h(n), the bias of is of orderh ^2km ,k=1, ...,t. Hereh ^m is the bias order of the ideal estimator (defined by using the unknownf as pilot). The second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates. Ifm<, it is possible to make asymptotically equivalent to the ideal estimator.     

Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.     

A reconstruction method for a two-dimensional inverse eigenvalue problem

In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.     

Weyl Pseudo-Differential Operator and Wigner Transform on the Poincaré Disk

The purpose of this paper is to investigate some relations between the kernel of a Weyl pseudo-differential operator and the Wigner transform on Poincaré disk defined in our previous paper [11]. The composition formula for the class of the operators defined in [11] has not been proved yet. However, some properties and relations, which are analogous to the Euclidean case, between the Weyl pseudo-differential operator and the Wigner transform have been investigated in [11]. In the present paper, an asymptotic formula for the Wigner transform of the kernel of a Weyl pseudo-differential operator as 0 is given. We also introduce a space of functions on the cotangent bundle T ^* D whose definition is based on the notion of the Schwartz space on the Poincaré disk. For an S ¹-invariant symbol in that space, we obtain a formula to reproduce the symbol from the kernel of the Weyl pseudo-differential operator.     

On the singular Bochner-Martinelli integral

The Bochner-Martinelli (B.-M.) kernel inherits, forn2, only some of properties of the Cauchy kernel in . For instance it is known that the singular B.-M. operatorM _n is not an involution forn2. M. Shapiro and N. Vasilevski found a formula forM ₂ ² using methods of quaternionic analysis which are essentially complex-twodimensional. The aim of this article is to present a formula forM _n ² for anyn2. We use now Clifford Analysis but forn=2 our formula coincides, of course, with the above-mentioned one.     

Zur Struktur von Quadraturformeln

Summary Usually the errorR _n (j) of a quadrature formula is estimated with the aid of theL ₁-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm Q _n of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing theL ₁-norm of the kernel. A remainder term of the typeR _n (f)=cf⁽ⁿ⁺¹⁾ () is possible iff the kernel is definite. In the case of an interpolatory formula this definiteness is usually shown by an application of the so-called V-method. We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh withm nodes there exists a rule with a small norm if the order is not greater than .     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.     

The problem of determining the one-dimensional kernel of the electroviscoelasticity equation

We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.     

Formulas for the Inverses of Toeplitz Matrices with Polynomially Singular Symbols

We consider large finite Toeplitz matrices with symbols of the form (1– cos )^p f() where p is a natural number and f is a sufficiently smooth positive function. By employing techniques based on the use of predictor polynomials, we derive exact and asymptotic formulas for the entries of the inverses of these matrices. We show in particular that asymptotically the inverse matrix mimics the Green kernel of a boundary value problem for the differential operator Submitted: June 20, 2003     

Orthogonal Decompositions in Hilbertian Subspaces, Error Functions and Optimal Extensions of Interpolation Systems

We address the question of optimal extensions of a fixed set S of measurement points for an interpolation system in a reproducing kernel Hilbert space of R . By considering the interpolation error function obtained from the reproducing kernel, we introduce different criteria of optimal extensions. We highlight the orthogonal decomposition of the space based on the subspace associated with the set S. The connection to the classical Schur decomposition of the Gram matrix of the interpolation problem is established. All the theory extends to conditionally positive functions and for this case, we give a result on the spectrum of the matrix of the interpolation problem.     

The principle of penalized empirical risk in severely ill-posed problems

We study a standard method of regularization by projections of the linear inverse problem Y=Ax+, where is a white Gaussian noise, and A is a known compact operator. It is assumed that the eigenvalues of AA^* converge to zero with exponential decay. Such behavior of the spectrum is typical for inverse problems related to elliptic differential operators. As model example we consider recovering of unknown boundary conditions in the Dirichlet problem for the Laplace equation on the unit disk. By using the singular value decomposition of A, we construct a projection estimator of x. The bandwidth of this estimator is chosen by a data-driven procedure based on the principle of minimization of penalized empirical risk. We provide non–asymptotic upper bounds for the mean square risk of this method and we show, in particular, that this approach gives asymptotically minimax estimators in our model example.Mathematics Subject Classification (2000):Primary 62G05, 62G20; secondary 62C20     

A hierarchically consistent,iterative sequence transformation

Recently, the author proposed a new nonlinear sequence transformation, the iterative transformation, which was shown to provide excellent results in several applications (Homeier [15]). In the present contribution, this sequence transformation is derived by a hierarchically consistent iteration of some basic transformation. Hierarchical consistency is proposed as an approach to control the well-known problem that the basic transformation can be generalized in many ways. Properties of the transformation are studied. It is of similar generality as the well-knownE algorithm (Brezinski [3], Håvie [18]). It is shown that the transformation can be implemented quite easily. In addition to the defining representation, there are alternative algorithms for its computation based on generalized differences. The kernel of the transformation is derived. The expression for the kernel is relatively compact and does not depend on any lower-order transforms. It is shown that several important other sequence transformations can be computed in an economical way using the transformation.Communicated by C. Brezinski     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Wavelet and Weyl Transforms Associated with the Spherical Mean Operator

The admissible wavelets associated with spherical mean operator and corresponding Weyl transforms are defined. The admissible condition is given in the generalized Fourier transforms, and Plancherel formula, Parseval formula, Reproducing formula and Reproducing kernel are studied. The criteria of boundedness of the Weyl transform on the L^p– spaces is given in term of the symbol function .     

Transformation Operator for Jacobi Matrices with Asymptotically Periodic Coefficients

We construct the transformation operator for the scattering problem with a periodic background under the assumption that the coefficients of the perturbation have a first finite moment. By means of the Marchenko approach [Marchenko, V. (1986) Sturm–Liouville Operators and Applications. Birkhäuser, Basel, Switzerland] we derive an estimate on the kernel of this transformation operator that allow us to study the inverse problem solution in the prescribed class of perturbations.     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017