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Linear algorithms of affine synthesis in the Lebesgue space \( L^1 [0,1]\). (English. Russian original) Zbl 1206.41007
Izv. Math. 74, No. 5, 993-1022 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 5, 115-144 (2010).
Systems of translates and dilates of integrable (\(L^1\)) functions (generators) over the unit interval are considered in this paper and, among other results, it is shown that they allow affine synthesis with respect to the model space of all absolutely summable coefficients vectors. For this, their integral over the unit interval must not vanish. In contrast to this, it is proved that there is no linear algorithm of affine synthesis in \(L^1\) for these systems of translates and dilates with respect to the space of all absolutely summable infinite coefficient sequences. However, if a normalisation condition on the generator is satisfied, and additionally a finiteness condition on an integral modulus of smoothness of the generator and a certain distance of it from the characteristic function of the unit interval holds, then again a linear algorithm is proved to exist. This is true as soon as the model space is replaced by the space of all coefficient sequences such that the series over the dilates and shifts of the generator with them converge with respect to \(L^1([0,1])\).

41A15 Spline approximation
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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