Approximation of functions of variable smoothness by Fourier-Legendre sums

Journal Article · Fri Jun 30 04:00:00 EDT 2000 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2000V191N05ABEH000480; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21202934

Sharapudinov, I I ^[1]

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1. Dagestan State University, Makhachkala (Russian Federation)

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}.

• OSTI ID: 21202934
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 5 Vol. 191; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
FOURIER ANALYSIS
LEGENDRE POLYNOMIALS
SMOOTH MANIFOLDS