Average Square Errors by Banach Measure of Recovery of Functions by Finite Sums of Terms of Their Trigonometric Fourier Series
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DOI:
https://doi.org/10.32523/bulmathenu.2025/1.2Keywords:
Fourier-Lebesgue trigonometric series, defining a function employing a complete set of its Fourier coefficients, probabilistic measurement on function classes, recovery of functions in average squareAbstract
The paper considers the problem of recovering functions using finite sums of terms of their trigonometric Fourier series with respect to probability measures on function classes characterized by the inability to determine a "spectrum of large coefficients". This difficulty motivates the consideration of arbitrary "finite sums of terms from the Fourier series".
The problem of defining probability measures on classes with individual estimates for trigonometric Fourier coefficients was resolved based on the fundamental nature of the property "A function can be defined in two ways: either by a rule or by the complete set of its trigonometric Fourier coefficients", as discussed in the monograph by V.M. Tikhomirov, The remaining steps were largely a matter of technical execution.
The transition to sequences of Fourier coefficients, using A.N. Kolmogorov’s extension theorem (on the extension of measures from finite-dimensional to infinite-dimensional spaces), allowed the introduction of a probability measure on classes with weighted Fourier coefficients, ultimately leading to the Banach measure – first constructed by Stefan Banach in the appendix to Stanisław Saks' book "Theory of the Integral". This paper also proposes some constructive details of the process of probabilistic measure
References
Suldin A.V. Mera Vinera i eye prilozheniya k priblizhennym metodam [Wiener’s measure and its appli- cations to approximate methods], Izv. VUZov[Izv. Universities]. Matematika-I: (6), 145-158(1959); II: (5), 165-179(1960).
Voronin S.M.. Skalyga V.I. O kvadraturnykh formulakh [On quadrature formulas], Dokl. AN SSSR [Dokl. USSR Academy of Sciences], 246(5), 1038-1041(1984).
Smale S. On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc., 13(2), 87-121(1985). DOI: https://doi.org/10.1090/S0273-0979-1985-15391-1
Traub J.F., Wasilkovski G.W, Wozniakovski H. Information-Based Complexity. (Academic Press, New York, 1988).
Temirgaliyev N. The concept of S.M.Voronin in the problem of comparisons of deterministic and random computation in the same terms, Bulletin of the L.N. Gumilyov Eurasian National University.Mathematics. Computer Science. Mechanics Series. 2019. Vol. 128. № 3. P. 8-33. DOI: https://doi.org/10.32523/2616-7182/2019-128-3-8-33
Banakh S. Integral Lebega v abstraktnom prostranstve [The Lebesgue integral in abstract space]// v kn. Saks S. Teoriya integrala [in the book. Saks S. Theory of the integral]. Moscow, IL. 1949. P. 463-477
Voronin S.M., Temirgaliev N. Application of Banach measure to quadrature formulas, Mathematical notes of the Academy of Sciences of the USSR, 39(1), 30-34(1986). DOI: https://doi.org/10.1007/BF01647628
Temirgaliyev N. O postroyenii veroyatnostnykh mer na funktsionalnykh klassakh[On the construction of prob- ability measures on functional classes], Trudy Matem. in-ta im. V.A. Steklova RAN [Proceedings of a math- ematical institute V.A. Steklov RAS.],218, 397-402(1997).
Kolmogorov A.N. Basic concepts of probability theory [Osnovnyye ponyatiya teorii veroyatnostey]. (Nauka, Moscow, 1974).
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