Abstract:
This paper is concerned with systems of functions on the unit interval which are generated by dyadic dilations and integer translations of a given function. Similar systems have a wide range of applications in the theory of wavelets, in nonlinear, and in particular, in greedy approximations, in the representation of functions by series, in problems in numerical analysis, and so on. Conditions, and in some particular cases, criteria for the generating function are given for the system to be Besselian, to form a Riesz basis or to be an orthonormal system, and separately, to be complete. For this purpose, the concept of the dual function of the generating function of a system is introduced and studied. Some of the conditions given below are easy to verify in practice, as is demonstrated by examples.
Bibliography: 25 titles.
Keywords:
Riesz basis, Haar system, affine system of functions, system of dilations and translations.
This research was carried out in the framework of the governmental target programme of the Ministry of Education and Science of the Russian Federation (project no 1.1520.2014/K) and with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00152-a).
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\by P.~A.~Terekhin
\paper Affine Riesz bases and the dual function
\jour Sb. Math.
\yr 2016
\vol 207
\issue 9
\pages 1287--1318
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Linking options:
https://www.mathnet.ru/eng/sm8221
https://doi.org/10.1070/SM8221
https://www.mathnet.ru/eng/sm/v207/i9/p111
This publication is cited in the following 5 articles:
Astashkin V S. Terekhin P.A.Y., “Sequences of Dilations and Translations Equivalent to the Haar System in l-P-Spaces”, J. Approx. Theory, 274 (2022), 105672
S. V. Astashkin, P. A. Terekhin, “Sequences of dilations and translations in function spaces”, J. Math. Anal. Appl., 457:1 (2018), 645–671
S. V. Astashkin, P. A. Terekhin, “Basis properties of affine Walsh systems in symmetric spaces”, Izv. Math., 82:3 (2018), 451–476
S. F. Lukomskii, P. A. Terekhin, S. A. Chumachenko, “Rademacher Chaoses in Problems of Constructing Spline Affine Systems”, Math. Notes, 103:6 (2018), 919–928
S. V. Astashkin, P. A. Terekhin, “Affine Walsh-type systems of functions in symmetric spaces”, Sb. Math., 209:4 (2018), 469–490
Citation in format AMSBIB
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