Approximation by Interpolation Trigonometric Polynomials in Metrics of the Space Lp on the Classes of Periodic Entire Functions
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 repres...
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| Published in: | Ukrainian mathematical journal Vol. 71; no. 2; pp. 322 - 332 |
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| Abstract | 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
→ ∞). |
|---|---|
| AbstractList | 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
→ ∞). |
| Author | Serdyuk, A. S. Sokolenko, I. V. |
| Author_xml | – sequence: 1 givenname: A. S. surname: Serdyuk fullname: Serdyuk, A. S. organization: Institute of Mathematics, Ukrainian National Academy of Sciences – sequence: 2 givenname: I. V. surname: Sokolenko fullname: Sokolenko, I. V. email: sokol@imath.kiev.ua organization: Institute of Mathematics, Ukrainian National Academy of Sciences |
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| References | A. S. Serdyuk, “Approximation of periodic analytic functions by interpolation trigonometric polynomials in the metric of the space L,” Ukr. Mat. Zh., 54, No. 5, 692–699 (2002); English translation: Ukr. Math. J., 54, No. 5, 852–861 (2002). A. S. Serdyuk, “Approximation of the classes of analytic functions by Fourier sums in uniform metric,” Ukr. Mat. Zh., 57, No. 8, 1079–1096 (2005); English translation: Ukr. Math. J., 57, No. 8, 1275–1296 (2005). A. S. Serdyuk, “Approximation of infinitely differentiable analytic functions by interpolation trigonometric polynomials in integral metric,” Ukr. Mat. Zh., 53, No. 12, 1654–1663 (2001); English translation: Ukr. Math. J., 53, No. 12, 2014–2026 (2001). A. S. Serdyuk and V. A. Voitovych, “Approximation of the classes of entire functions by interpolation analogs of the de la Vall´ee- Poussin sums,” in: Approximation Theory of Functions and Related Problems [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 7, No. 1 (2010), pp. 274–297. StepanetsA. I.SerdyukA. S.ShidlichA. L.Classification of infinitely differentiable periodic functionsUkrainian Mathematical Journal2008601219822005252311610.1007/s11253-009-0185-1 R. B. Stechkin, “Estimation of the remainder of Fourier series for differentiable functions,” in: Approximation of Functions by Polynomials and Splines, Tr. Mat. Inst. Akad. Nauk SSSR, 145, 126–151 (1980). KorneichukNPExact Constants in Approximation Theory [in Russian]1987MoscowNauka A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002). A. S. Serdyuk, “Approximation of periodic functions of high smoothness by interpolation trigonometric polynomials in the metric of L1,” Ukr. Mat. Zh., 52, No. 7, 994–998 (2000); English translation: Ukr. Math. J., 52, No. 7, 1141–1146 (2000). SharapudinovIIOn the best approximation and polynomials of the least quadratic deviationAnal. Math.19839322323473298410.1007/BF01989807 OskolkovKIInequalities of the ‘large sieve’ type and applications to problems of trigonometric approximationAnal. Math.198612214316685453610.1007/BF02027298 TelyakovskiiS. A.Approximation of functions of higher smoothness by Fourier sumsUkrainian Mathematical Journal1989414444451100485710.1007/BF01060623 V. P. Motornyi, “Approximation of periodic functions by interpolation polynomials in L1,” Ukr. Mat. Zh., 42, No. 6, 781–786 (1990); English translation: Ukr. Math. J., 42, No. 6, 690–693 (1990). II Sharapudinov (1647_CR3) 1983; 9 1647_CR9 NP Korneichuk (1647_CR10) 1987 KI Oskolkov (1647_CR4) 1986; 12 A. I. Stepanets (1647_CR2) 2008; 60 1647_CR1 S. A. Telyakovskii (1647_CR13) 1989; 41 1647_CR12 cr-split#-1647_CR7.2 cr-split#-1647_CR8.1 cr-split#-1647_CR11.1 cr-split#-1647_CR6.2 cr-split#-1647_CR7.1 cr-split#-1647_CR5.2 cr-split#-1647_CR6.1 cr-split#-1647_CR5.1 cr-split#-1647_CR8.2 cr-split#-1647_CR11.2 |
| References_xml | – ident: #cr-split#-1647_CR7.1 – ident: 1647_CR9 – ident: 1647_CR12 – volume: 60 start-page: 1982 issue: 12 year: 2008 ident: 1647_CR2 publication-title: Ukrainian Mathematical Journal doi: 10.1007/s11253-009-0185-1 contributor: fullname: A. I. Stepanets – ident: #cr-split#-1647_CR5.2 – ident: #cr-split#-1647_CR5.1 – volume: 9 start-page: 223 issue: 3 year: 1983 ident: 1647_CR3 publication-title: Anal. Math. doi: 10.1007/BF01989807 contributor: fullname: II Sharapudinov – ident: #cr-split#-1647_CR7.2 – ident: #cr-split#-1647_CR11.1 – ident: #cr-split#-1647_CR11.2 – ident: 1647_CR1 – volume: 41 start-page: 444 issue: 4 year: 1989 ident: 1647_CR13 publication-title: Ukrainian Mathematical Journal doi: 10.1007/BF01060623 contributor: fullname: S. A. Telyakovskii – ident: #cr-split#-1647_CR8.1 – ident: #cr-split#-1647_CR8.2 – volume-title: Exact Constants in Approximation Theory [in Russian] year: 1987 ident: 1647_CR10 contributor: fullname: NP Korneichuk – volume: 12 start-page: 143 issue: 2 year: 1986 ident: 1647_CR4 publication-title: Anal. Math. doi: 10.1007/BF02027298 contributor: fullname: KI Oskolkov – ident: #cr-split#-1647_CR6.1 – ident: #cr-split#-1647_CR6.2 |
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| SubjectTerms | Algebra Analysis Applications of Mathematics Geometry Mathematics Mathematics and Statistics Statistics |
| Title | Approximation by Interpolation Trigonometric Polynomials in Metrics of the Space Lp on the Classes of Periodic Entire Functions |
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