Abstract
Along with the convergence in \(L^2\)-norm, convergence almost everywhere of expansions in functional systems is a property of interest for both theoretical studies and applications. In this paper we present results on convergence almost everywhere for orthorecursive expansions which are a natural generalization of classical expansions in orthogonal systems. As a corollary of a more general result, we obtain a condition on coefficients of an expansion that guarantees convergence almost everywhere. We also show that this condition cannot be relaxed.
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References
Galatenko, V.V.: On the orthorecursive expansion with respect to a certain function system with computational errors in the coefficients. Mat. Sb. 195(7), 935–949 (2004)
Galatenko, V.V.: On orthorecursive expansions with errors in the calculation of coefficients. Izv. Math. 69(1), 1–14 (2005)
Kashin, B.S., Saakyan, A.A.: Orthogonal Series. American Mathematical Society, Providence (1989)
Knopp, K.: Theory and Applications of Infinite Series. Dover Publications Inc., New York (1990)
Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications Inc., New York (1975)
Kudryavtsev, A.Yu.: On the convergence of orthorecursive expansions in nonorthogonal wavelets. Math. Notes. 92(5), 643–656 (2012)
Kudryavtsev, A.Yu.: On the rate of convergence of orthorecursive expansions over non-orthogonal wavelets. Izv. Math. 76(4), 688–701 (2012)
Lukashenko, T.P.: Properties of orthorecursive expansions in nonorthogonal systems. Moscow Univ. Math. Bull. 56(1), 5–9 (2001)
Men’shov, D.E.: Sur les series de fonctions orthogonalen I. Fund. Math. 4, 82–105 (1923)
Politov, A.V.: A convergence criterion for orthorecursive expansions in Euclidean spaces. Math. Notes. 93(3), 636–640 (2013)
Acknowledgments
The authors thank Dr. Alexey Galatenko for valuable comments and discussions.
The research was supported by the Russian Foundation for Basic Research (project 14–01–00417) and the President grant for the support of the leading scientific schools of the Russian Federation (grant NSh–7461.2016.1).
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Galatenko, V.V., Lukashenko, T.P., Sadovnichiy, V.A. (2016). Convergence Almost Everywhere of Orthorecursive Expansions in Functional Systems. In: Sadovnichiy, V., Zgurovsky, M. (eds) Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-40673-2_1
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DOI: https://doi.org/10.1007/978-3-319-40673-2_1
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