Sharp Constant in Jackson’s Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions
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Date
2012
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Pleiades Publishing
Abstract
EN: Abstract—It is proved that, in the space C2π, for all k,n ∈ N, n > 1, the following inequalities hold:
1 − 1
2n
k2 + 1
2 ≤ sup
f∈C2π
f=const
en−1(f)
ω2(f,π/(2nk)) ≤
k2 + 1
2 .
where en−1(f) is the value of the best approximation of f by trigonometric polynomials and ω2(f,h)
is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous
polygonal lines with equidistant nodes.
Description
S. Pichugov: ORCID 0000-0002-4263-4429
Keywords
Jackson’s inequality, periodic function, trigonometric polynomial, modulus of smoothness, polygonal line, Steklov mean, Favard sum, КПМ
Citation
Pichugov S. A. Sharp Constant in Jackson’s Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions / S. A. Pichugov // Mathematical Notes. – 2013. – Vol. 93, № 6. – Р. 116-121. – DOI: 10.1134/S000143461305011.
Pichugov S. A. Sharp Constant in Jackson’s Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions. Mathematical Notes. Vol. 93, No. 5-6. P. 917–922. DOI: 10.1134/S0001434618110421. (Russian original Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 932–938). Full text.
Pichugov S. A. Sharp Constant in Jackson’s Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions. Mathematical Notes. Vol. 93, No. 5-6. P. 917–922. DOI: 10.1134/S0001434618110421. (Russian original Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 932–938). Full text.