A general theorem on generation of moments-preserving cosine families by Laplace operators in C [0, 1] (original) (raw)

Generation of cosine families via Lord Kelvin’s method of images

Journal of Evolution Equations, 2010

We show that generation theorems for cosine families related to one-dimensional Laplacians in C[0, ∞] may be obtained by Lord Kelvin's method of images, linking them with existence of invariant subspaces of the basic cosine family. This allows us to deal with boundary conditions more general than those considered before (Bátkal and Engel in to give explicit formulae for transition kernels of related Brownian motions on [0, ∞). As another application we exhibit an example of a family of equibounded cosine operator functions in C[0, ∞] that converge merely on C 0 (0, ∞] while the corresponding semigroups converge on the whole of C[0, ∞].

Some New Aspects of the L-Moment Problem

2010

This note is devoted to the L-moment problem. The L-moment problem consists of characterising the sequence of moments an = ∫ Rt nf(t)dt, n ∈ N of a real measurable function f (with prescribed support) which satisfies a condition such is 0 ≤ f ≤ L a.e. dt. The L-moment problem was formulated and completely solved by Akhiezer and Krein [2] in the thirties. The interest for the moments of a bounded measurable function on the real axis goes back to A.A. Markov in the last part of the nineteenth century. It was M.G. Krein who studied again this field with new methods at that time of function theory and functional analysis. The aim of the present paper is to study a two dimensional complex L-moment problem. The L-complex moment problem consists of characterising a double complex sequence {yα,β}α,β∈N to represent the moments of a measurable real function h defined on the closed unit disc of the complex plane, which satisfies a condition of boundness 0 ≤ h ≤ L for a positive constant L >...

Operator-valued trigonometric moment and L-moment problems

2009

We give a necessary and sufficient condition for a sequence of complex matrices to be represented as a trigonometric moment sequence. We also give a necessary and sufficient condition for a sequence of complex matrices to generate L-scalar moment sequences with respect to real-valued measurable functions.

Generation of Cosine Families on L p (0,1) by Elliptic Operators with Robin Boundary Conditions

Functional Analysis and Evolution Equations, 2007

Let a ∈ W 1,∞ (0, 1), a(x) ≥ α > 0, b, c ∈ L ∞ (0, 1) and consider the differential operator A given by Au = au + bu + cu. Let α j , β j (j = 0, 1) be complex numbers satisfying (α j , β j ) = (0, 0) for j = 0, 1. We prove that a realization of A with the boundary conditions α j u (j) + β j u(j) = 0, j = 0, 1, generates a cosine family on L p (0, 1) for every p ∈ [1, ∞). This result is obtained by an explicit calculation, using simply d'Alembert's formula, of the solutions in the case of the Laplace operator.

On the truncated operator trigonometric moment problem

Concrete Operators, 2015

In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.

On Certain Operator Families Related to Cosine Operator Functions

Taiwanese Journal of Mathematics, 1997

This paper is concerned with two cosine-function-related functions which are called cosine step response and cosine cumulative output. We study some of their properties, such as measurability, continuity, infinitesimal operator, compactness, positivity, almost periodicity, and asymptotic behavior.

On the moment problem in the bounded case

Journal of Computational and Applied Mathematics, 1993

Torrano, E. and R. Guadalupe, On the moment problem in the bounded case, Journal of Computational and Applied Mathematics 49 Through the matrix treatment of the theory of orthogonal polynomials on curves or domains of the complex plane, we extend to arbitrary bounded regions the results of for the unit disc.

On the convergence and approximation of cosine functions

Aequationes Mathematicae, 1974

201 1. Let X be a Banach space with norm II" It. M (X) denotes the space of all bounded linear operators on X. A cosine function is afamily of operators C=(C(t):teR= (-~, ~)) ~(X) satisfying (i) C(t+s)+C(t-s)=2C(t) C(s) for all t, seR, (ii) C (0) = I (= the identity operator), (iii) C(-)x: R ~ X is (strongly) continuous for each x~X. The (infinitesimal) generator A of a cosine function C is the operator A = C" (0). Here ' means differentiation with respect to t. The domain of A is D (A) = {xeX: the second strong derivative C"(t)x exists at t=0). Cosine functions are important since they bear the same relation to second order Cauchy problems as do semigroups to first order Cauchy problems. Specifically, the Cauchy problem for u" (t) = Au (t) (t ~ R) is well-posed if and only if A is the generator of a cosine function. (Cf. Fattorini [2, 3] for a precise formulation of this and for further results along this line.) The following generation theorem, due to Sova [9], DaPrato and Giusti [1], and Fattorini [2], is the cosine function analog of the Hille-Yosida semigroup generation theorem. THEOREM O. A is the generator of a cosine function C if and only if A is closed, densely defined, and there are constants M>0, to>~0 such that for 1>to, 22 is in the resolvent set of A and II (am/dim) [2 (22/-A)-1] II <~ Mm ! (2-co)-'~-i for all m~Z + = {0, 1, 2,...}. In this case, IfC(t)ll ~< Me ~'N for all teR, and cJO 2(221-a)-'x = f e-arC(t) xdt itl o for all 1>co and x~X.

Approximation properties for linear combinations of moment type operators

Computers & Mathematics with Applications, 2011

where (K n ) n is a kernel satisfying suitable assumptions and f belongs to L p -spaces. These operators were extensively studied by Butzer and Jansche in in connection with the Mellin transform theory. Among the above operators, an important example is given by the moment (or average) operator whose kernel is given by (see Section 2)