We study the theory of spectral measures in topological vector spaces. We extend the Hilbert space theory to this setting and generalize the notion of spectral measures in some useful ways to provide a framework for operator theory in this setting. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (...

For Machine Learning (ML) classification problem, where a vector of x–observations (values of attributes) is mapped to a single y value (class label), a generalized Radon– Nikodym type of solution is proposed. Quantum–mechanics –like probability states ψ2(x) are considered and “Cluster Centers”, corresponding to the extremums of < yψ2(x) > / < ψ2(x) >, are found from generalized eigenvalues pro...

Let σ̂ be a Cauchy transform of a possibly complex-valued Borel measure σ and {pn} be a system of orthonormal polynomials with respect to a measure μ, supp(μ)∩ supp(σ) = ∅. An (m,n)-th Frobenius-Padé approximant to σ̂ is a rational function P/Q, deg(P) 6m, deg(Q) 6 n, such that the first m+n+ 1 Fourier coefficients of the linear form Qσ̂−P vanish when the form is developed into a series with respe...

This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum integral’s form for simple functions is characterized and it is shown that the quantum integral generalizes the Lebesgue integral. A bounded, monotone convergenc...

The aim of these notes is to illustrate a proof of the following remarkable Theorem of Alberti (first proved in [1]). Here, when μ is a Radon measure on Ω ⊂ R, we denote by μ its absolutely continuous part (with respect to the Lebesgue measure L ), by μ := μ− μ its singular part, and by |μ| its total variation measure. Clearly, |μ|a = |μa| and |μ|s = μ. When μ = Du for some u ∈ BV (Ω,R), we wil...