A Hilbert space approach to singularities of functions

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We introduce the notion of a pseudomultiplier of a Hilbert space H of functions on a set Ω. Roughly, a pseudomultiplier of H is a function which multiplies a finite-codimensional subspace of H into H, where we allow the possibility that a pseudomultiplier is not defined on all of Ω. A pseudomultiplier of H has singularities, which comprise a subspace of H, and generalize the concept of singularities of an analytic function, even though the elements of 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.