Systems theory: A hilbert space approach

A Functional Hilbert Space Approach to the Theory of Wavelets

We approach the theory of wavelets from the theory of functional Hilbert spaces. Starting with a Hilbert space H, we consider a subset V of H, for which the span is dense in H. We define a function of positive type on the index set I which labels the elements of V . This function of positive type induces uniquely a functional Hilbert space, which is a subspace of CI and there exists a unitary m...

Elementary Hilbert Space Theory

A Hilbert Space is an inner product space that is complete. Let H be a Hilbert space and for f, g ∈ H, let 〈f, g〉 be the inner product and ‖f‖ = √ 〈f, f〉. (1.1) We will consider Hilbert spaces over C. Most arguments go through for Hilbert spaces over R, and the arguments are simpler. We assume that the reader is familiar with some of the basic facts about the inner product. Let us prove a few t...

A Hilbert space approach to self-similar systems

This paper investigates the structural properties of linear self-similar systems, using an invariant subspace approach. The self-similar property is interpreted in terms of invariance of the corresponding transfer function space to a given transformation in a Hilbert space, in a same way as the time invariance property for linear systems is related to the shift-invariance of the Hardy spaces. T...

OPERATOR THEORY ON HILBERT SPACE Class notes

intersection theory and operators in Hilbert space

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial zeros of the Riemann zeta-function arise from an operator of this class, the original Riemann Hypothesis is equivalent to the existence of an abstract inter...