A Hilbert space approach to singularities of functions

TL;DR

This paper introduces pseudomultipliers in Hilbert spaces of functions, analyzing their singularities and providing a classification that generalizes the concept of singularities of analytic functions without requiring analyticity.

Contribution

It defines pseudomultipliers in Hilbert spaces, explores their singularities, and offers a broad classification in function-theoretic terms, extending classical notions beyond analytic functions.

Findings

Pseudomultipliers can be defined as functions multiplying a finite-codimensional subspace into the Hilbert space.

Singularities of pseudomultipliers form a subspace of the Hilbert space.

A broad classification of these singularities is provided in function-theoretic terms.

Abstract

We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal H$ of functions on a set $\Omega$. Roughly, a pseudomultiplier of $\mathcal H$ is a function which multiplies a finite-codimensional subspace of $\mathcal H$ into $\mathcal H$, where we allow the possibility that a pseudomultiplier is not defined on all of $\Omega$. A pseudomultiplier of $\mathcal H$ has singularities, which comprise a subspace of $\mathcal H$, and generalize the concept of singularities of an analytic function, even though the elements of $\mathcal H$ need not enjoy any sort of analyticity. We analyse the natures of these singularities, and obtain a broad classification of them in function-theoretic terms.