Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous functiong ₁ εC ₀[0,1]² with support in the rectangle [0,1]×[0,1/2] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1]×[1/2,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.     

Multiresolution weighted norm equivalences and applications

Summary. We establish multiresolution norm equivalences in weighted spaces L ² _w ((0,1)) with possibly singular weight functions w(x)≥0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w(x) within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning p-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from finance. Revised version received March 19, 2003 Mathematics Subject Classification (2000): 65F35, 65F50, 65N22, 65N35, 65N30, 65T60, 60H10, 60H35 An erratum to this article is available at .     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g ₁∈C ₀ [0,1]² with support in the rectangle [0,1] × [0,?] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1] × [?,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer. Received: 21 December 1995 / Revised version: 5 October 1996     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

Exponential stability for a one‐dimensional compressible viscous micropolar fluid

In this paper, we consider one‐dimensional compressible viscous and heat‐conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (H¹(0,1))⁴, and then we establish the global existence and exponential stability of solutions in (H²(0,1))⁴ under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

Existence and uniqueness for a nonlinear inverse reaction‐diffusion problem with a nonlinear source in higher dimensions

This paper analyzes the existence and the uniqueness problem for an n‐dimensional nonlinear inverse reaction‐diffusion problem with a nonlinear source. A transformation is used to obtain a new inverse coefficient problem. Then, a parabolic differential operator L_λ is defined to establish the relation between the solution of L_λ = 0 and the new inverse problem. Following this, it is shown that the inverse problem has at least one solution in the class of admissible coefficients. Furthermore, it is proved that this solution is the unique solution of the undertaken inverse problem. A numerical example is given to illustrate ill‐posedness of the inverse problem. Copyright © 2013 John Wiley & Sons, Ltd.     

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besov spaces B _r,r ⁶ (0.1) with 0<σ<∞ and (1+σ)⁻¹相似文献   

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Linear versus Non-Linear Acquisition of Step-Functions

We address in this article the following two closely related problems. 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way. 2. How to reconstruct such functions from a small number of measurements. The stress is on a comparison of linear and non-linear approaches. As a model case, we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function ℋ _t =χ _[0,t]. Considered as a curve in the Hilbert space L ²([0,1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next, we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov’s n-width and ε-entropy, as well as Temlyakov’s (N,m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other. This research was supported by the ISF, Grant No. 304/05, and by the Minerva Foundation.     

Closed ideals in with the Duhamel product as multiplication

Let (C^∞,) denote the algebra of infinitely differentiable functions in [0,1] with Duhamel product as multiplication. We describe all the closed ideals in (C^∞,). As a consequence we obtain that the integration operator I, , is unicellular in the space C^∞[0,1], which is the solution of a long-standing problem.     

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

The object of this paper is threefold. First, we investigate in a Hilbert space setting the utility of approximate source conditions in the method of Tikhonov–Phillips regularization for linear ill‐posed operator equations. We introduce distance functions measuring the violation of canonical source conditions and derive convergence rates for regularized solutions based on those functions. Moreover, such distance functions are verified for simple multiplication operators in L²(0, 1). The second aim of this paper is to emphasize that multiplication operators play some interesting role in inverse problem theory. In this context, we give examples of non‐linear inverse problems in natural sciences and stochastic finance that can be written as non‐linear operator equations in L²(0, 1), for which the forward operator is a composition of a linear integration operator and a non‐linear superposition operator. The Fréchet derivative of such a forward operator is a composition of a compact integration and a non‐compact multiplication operator. If the multiplier function defining the multiplication operator has zeros, then for the linearization an additional ill‐posedness factor arises. By considering the structure of canonical source conditions for the linearized problem it could be expected that different decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely different ill‐posedness situations. As third we apply the results on approximate source conditions to such composite linear problems in L²(0, 1) and indicate that only integrals of multiplier functions and not the specific character of the decay of multiplier functions in a neighbourhood of a zero determine the convergence behaviour of regularized solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of R_II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

hp-TDG/IPDG for Parabolic Obstacle Problems

We present a mixed hp-FE time DG method combined with an interior penalty DG method for parabolic obstacle problems. The discrete Lagragian multiplier set is spanned by basis functions which are biorthogonal to the basis functions of the primal variable. This allows us to write the discrete mixed problem as a problem of finding the root of a strongly semi-smooth function. In turn, this problem is solved by a locally Q-quadratic converging semi-smooth Newton method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

The boundary-value problem for Lavrent’Ev–Bitsadze equation with two internal lines of change of a type

We study the problem with boundary conditions of the first and second kind on the boundary of a rectangular domain for an equation with two internal perpendicular lines of change of a type. With the use of the spectral method we prove the unique solvability of the mentioned problem. The eigenvalue problem for an ordinary differential equation obtained by separation of variables is not self-adjoint, and the system of root functions is not orthogonal. We construct the corresponding biorthogonal system of functions and prove its completeness. This allows us to establish a criterion for the uniqueness of the solution to the problem under consideration. We construct the solution as the sum of the biorthogonal series.