Finite Convolution Operators Research Articles

A Characterization of the range of a finite convolution operator with a hankel kernel

We consider the operator M of convolving a function g supported in the finite interval [-1,1] with the Hankel function . We derive a characterization of the range of M, considered as an operator on LP ([—1,1]), in terms of the Hilbert transform. This characterization can be used as a starting point for solving the integral equation Mg=f with f contained in the range of M. The characterization of the range of M is obtained by applying contour integration methods in the complex plane and it can be expected that by applying such techniques also the ranges of other integral operators can be characterized

Semigroups of finite convolution operators

Semigroups of operators in the commutant A of the Volterra operator J: ƒ(x) → ∝ 0 x f(t) dt on L 2 (0, 1) are constructed and studied by means of a complex Fourier transform technique. A sufficient condition for a fixed operator in the algebra A to be embeddable in a C 0 semigroup of operators in A is given. The condition is shown to be necessary under the further restriction that the semigroup consists of contraction operators (see Theorem 3). In this case, a description of all C 0 semigroups of operators in A containing the fixed operator is obtained (see Theorem 2). Under mild conditions, the domain of analyticity of a semigroup of operators in A is shown to be a sector. Relationships between the lattices of closed invariant subspaces of the operators comprising a semigroup of operators in A and the resolvents of the infinitesimal generator of the semigroup are studied.

Invariant subspaces of direct sums of finite convolution operators

In general, a direct sum of finite convolution operators has a very complicated invariant subspace structure. In some cases, however, the invariant subspaces split and can be exactly described. This phenomenon is studied with the aid of a theorem that relates lattices of invariant subspaces and weakly closed algebras of operators.

Spectral analysis of finite convolution operators with matrix kernels

Let N be a finite dimensional complex Hilbert space. A finite convolution operator on the vector function space L N 2 (0,1) is an operator T of the form (Tf) (x) = ∫ 0 x k (x−t)f(t)dt, where k(t) is a norm integrableB(N)-valued function on [0,1]. A symbol for T is any function A(z) of the form A(z) = ∫ 0 1 k(t)eitzdt + eizG(z), where G(z) is aB(N)-valued function which is bounded and analytic in a half plane y > η. It is shown that under suitable restrictions two finite convolution operators are similar if their symbols are asymptotically close as z → ∞ in a half plane y > η .

Finite convolution operators

Finite convolution operators

Spectral analysis of finite convolution operators

In this paper the similarity problem for operators of the form ( ∗ ) T : f ( x ) → ∫ 0 x k ( x − t ) f ( t ) d t ( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt on L 2 ( 0 , 1 ) {L^2}(0,1) is studied. Let K ( z ) = ∫ 0 1 k ( t ) e i t z d t K(z) = \smallint _0^1\;k(t){e^{itz}}dt . A function C ( z ) C(z) is called a symbol for T if C ( z ) C(z) can be written in the form C ( z ) = K ( z ) + e i z G ( z ) C(z) = K(z) + {e^{iz}}G(z) , where G ( z ) G(z) is a function bounded and analytic in a half plane y > δ y > \delta , for some real number δ \delta . Under suitable restrictions, it is shown that two operators of the form ( ∗ ) ( \ast ) will be similar if they possess symbols which are asymptotically close together as z → ∞ z \to \infty in some half plane y > δ y > \delta .

A Note on Finite Convolution Operators

Integral operators having a kernel of convolution type on a finite interval play an important part in noise theory, in the representation of band-limited signals and in other parts of applied mathematics [1]. As such, it would seem important to investigate the properties they possess beyond those that they inherit from the Fredholm and Hilbert-Schmidt theories. This note arose from an (unpublished) observation of H. 0. Pollak in his studies of the Dirichlet kernel occurring in the theory of band-limited functions. The observation was, in effect, that if the operator-theoretic square of a finite convolution operator (defined below) were again one, then the eigenfunctions are exponentials. He asked whether a similar result held for more general functions of an operator, or When can two such operators commute? We prove a theorem below which gives a partial answer. A finite convolution operator is an integral operator of the form