Approximation of functions of variable smoothness by Fourier-Legendre sums
Abstract
Assume that 0<{mu}{<=}1, and let r{>=}1 be an integer. Let {delta}={l_brace}a{sub 1},...,a{sub l}{r_brace}, where the a{sub i} are points in the interval (-1,1). The classes S{sup r}H{sup {mu}}{sub {delta}} and S{sup r}H{sup {mu}}{sub {delta}}(B) are introduced. These consist of functions with absolutely continuous (r-1)th derivative on [-1,1] such that their rth and (r+1)th derivatives satisfy certain conditions outside the set {delta}. It is proved that for 0<{mu}<1 the Fourier-Legendre sums realize the best approximation in the classes S{sup r}H{sup {mu}}{sub {delta}}(B). Using the Fourier-Legendre expansions, polynomials Y{sub n+2r} of order n+2r are constructed that possess the following property: for 0<{mu}<1 the {nu}th derivative of the polynomial Y{sub n+2r} approximates f{sup ({nu})}(x) (f element of S{sup r}H{sup {mu}}{sub {delta}}) on [-1,1] to within O(n{sup {nu}}{sup +1-r-{mu}}), and the accuracy is of order O(n{sup {nu}}{sup -r-{mu}}) outside {delta}.
- Authors:
-
- Dagestan State University, Makhachkala (Russian Federation)
- Publication Date:
- OSTI Identifier:
- 21202934
- Resource Type:
- Journal Article
- Journal Name:
- Sbornik. Mathematics
- Additional Journal Information:
- Journal Volume: 191; Journal Issue: 5; Other Information: DOI: 10.1070/SM2000v191n05ABEH000480; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; FOURIER ANALYSIS; LEGENDRE POLYNOMIALS; SMOOTH MANIFOLDS
Citation Formats
Sharapudinov, I I. Approximation of functions of variable smoothness by Fourier-Legendre sums. United States: N. p., 2000.
Web. doi:10.1070/SM2000V191N05ABEH000480; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Sharapudinov, I I. Approximation of functions of variable smoothness by Fourier-Legendre sums. United States. https://doi.org/10.1070/SM2000V191N05ABEH000480; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)
Sharapudinov, I I. 2000.
"Approximation of functions of variable smoothness by Fourier-Legendre sums". United States. https://doi.org/10.1070/SM2000V191N05ABEH000480; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21202934,
title = {Approximation of functions of variable smoothness by Fourier-Legendre sums},
author = {Sharapudinov, I I},
abstractNote = {Assume that 0<{mu}{<=}1, and let r{>=}1 be an integer. Let {delta}={l_brace}a{sub 1},...,a{sub l}{r_brace}, where the a{sub i} are points in the interval (-1,1). The classes S{sup r}H{sup {mu}}{sub {delta}} and S{sup r}H{sup {mu}}{sub {delta}}(B) are introduced. These consist of functions with absolutely continuous (r-1)th derivative on [-1,1] such that their rth and (r+1)th derivatives satisfy certain conditions outside the set {delta}. It is proved that for 0<{mu}<1 the Fourier-Legendre sums realize the best approximation in the classes S{sup r}H{sup {mu}}{sub {delta}}(B). Using the Fourier-Legendre expansions, polynomials Y{sub n+2r} of order n+2r are constructed that possess the following property: for 0<{mu}<1 the {nu}th derivative of the polynomial Y{sub n+2r} approximates f{sup ({nu})}(x) (f element of S{sup r}H{sup {mu}}{sub {delta}}) on [-1,1] to within O(n{sup {nu}}{sup +1-r-{mu}}), and the accuracy is of order O(n{sup {nu}}{sup -r-{mu}}) outside {delta}.},
doi = {10.1070/SM2000V191N05ABEH000480; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
url = {https://www.osti.gov/biblio/21202934},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 5,
volume = 191,
place = {United States},
year = {Fri Jun 30 00:00:00 EDT 2000},
month = {Fri Jun 30 00:00:00 EDT 2000}
}