Approximation by Interpolation Trigonometric Polynomials in Metrics of the Space Lp on the Classes of Periodic Entire Functions

• Published in: Ukrainian mathematical journal Vol. 71; no. 2; pp. 322 - 332
• Main Authors: Serdyuk, A. S., Sokolenko, I. V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-07-2019
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-019-01647-2

We establish asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with equidistant distribution of interpolation nodes
x
k
n
−
1
=
2
kπ
2
n
−
1
,
k
∈
ℤ
,
in metrics of the spaces
L
p
on the classes of 2
⇡
-periodic functions that can be represented in the form of convolutions of functions 𝜑
,
𝜑 ⊥ 1
,
from the unit ball in the space
L
1
with fixed generating kernels in the case where the modules of their Fourier coefficients
ψ
(
k
) satisfy the condition lim
k
 → ∞
ψ
(
k
 + 1)/
ψ
(
k
) = 0. Similar estimates are also obtained for the classes of
r
–differentiable functions
W
1
r
with rapidly increasing exponents of smoothness
r
(
r
/
n
 → ∞, 
n
 → ∞).