Abstract. Let h be a generalized frame in a separable Hilbert space H indexed by a measure space (M, S, µ), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the corresponding index measure space (M, S, µ) can be decomposed into atoms E 1 , E 2 , · · · such that L 2 (µ) is isometrically isomorphic to the weighted space 2 w of all sequences {c i } of complex numbers with ||{c i }|| 2 = |c i | 2 w i < ∞, whereThis provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in L 2 (R).
We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0 < α < 1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α → 1 − we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
Let A and B be nonempty subsets of a metric space X and also T :for some > 0. We call pair A, B an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, T -minimizing a sequence are given in a metric space.
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