Abstract
General Rogosinsky–Bernstein linear polynomial means \(R_n(f)\) of Fourier series are introduced and three convergence criteria as \(n\to \infty\) are obtained: for convergence in the space \(C\) of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and \(d\)-points). For smooth functions, the rate of convergence in norm of \(R_n(f)\), as well as of their interpolation analogues, is also studied. For approximation of functions in \(C^r\), the asymptotics is found along with the rate of decrease of the remainder term.
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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 592-605 https://doi.org/10.4213/mzm13450.
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Trigub, R.M. Rogosinsky–Bernstein Polynomial Method of Summation of Trigonometric Fourier Series. Math Notes 111, 604–615 (2022). https://doi.org/10.1134/S0001434622030294
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DOI: https://doi.org/10.1134/S0001434622030294
Keywords
- series and Fourier transforms
- Hardy’s inequality
- Riesz means
- Lebesgue points (\(l\)-points) and \(d\)-points
- modulus of smoothness
- linearized modulus of smoothness
- Jackson’s theorem
- Vallée-Poussin polynomial
- conjugate function
- entire functions of exponential type
- comparison principle
- Marcinkiewicz’s inequality and discretization