Eigenfunction Expansions and Transformation Theory

Eigenfunction Expansions and Transformation Theory

Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space Φ× in a convenient Gelfand triplet Φ ⊆ H ⊆ Φ×. This work presents a fit treatment for computational purposes of transformations formulas relating different generalized bases of eigenfunctions in both frameworks direct integrals and Gelfand tripl...

q-Sturm-Liouville theory and the corresponding eigenfunction expansions

The aim of this paper is to study the q-Schrödinger operator L = q(x) −∆q, where q(x) is a given function of x defined over R q = {qn, n ∈ Z} and ∆q is the q-Laplace operator ∆qf(x) = 1 x [ f(qx)− 1 + q q f(x) + 1 q f(qx) ]

Perturbation of eigenfunction expansions.

Convergence of Generalized Eigenfunction Expansions

We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperboli...

Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifold by Seeley [See69], and the characterisation of Gevrey functions and Gevrey u...