Polynomial approximation of a function and its first derivative in near minimax norms

Author:
Edgar A. Cohen

Journal:
Math. Comp. **27** (1973), 817-827

MSC:
Primary 41A10

MathSciNet review:
0330843

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Abstract: Two near minimax norms for polynomial approximation are presented. They are designed for approximation of both a function and its first derivative uniformly by polynomials over a given finite interval. The first one is a convex combination of two integrals, one involving the function and the other the derivative, and the second is the sum of the square of the value of the function at one point and an integral involving the derivative. For any smooth function defined on a finite closed interval, one forms a generalized Chebyshev polynomial expansion to approximate both the function and derivative uniformly.

**[1]**E. K. Blum and P. C. Curtis Jr.,*Asymptotic behavior of the best polynomial approximation*, J. Assoc. Comput. Mach.**8**(1961), 645–647. MR**0130522****[2]**C. W. Clenshaw and A. R. Curtis,*A method for numerical integration on an automatic computer*, Numer. Math.**2**(1960), 197–205. MR**0117885****[3]**Edgar A. Cohen Jr.,*Theoretical properties of best polynomial approximation in 𝑊^{1,}²[-1,1]*, SIAM J. Math. Anal.**2**(1971), 187–192. MR**0294963****[4]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[5]**Siegfried Filippi,*Angenäherte Tschebyscheff-Approximation einer Stammfunktion—eine Modifikation des Verfahrens von Clenshaw und Curtis*, Numer. Math.**6**(1964), 320–328 (German). MR**0170472****[6]**A. Meir and A. Sharma,*Simultaneous approximation of a function and its derivatives*, SIAM J. Numer. Anal.**3**(1966), 553–563. MR**0216209****[7]**D. G. Moursund,*Chebyshev approximations of a function and its derivatives*, Math. Comp.**18**(1964), 382–389. MR**0166529**, 10.1090/S0025-5718-1964-0166529-5**[8]**I. P. Natanson,*Teoriya funkciĭ veščestvennoĭ peremennoĭ*, Gosudarstv. Izdat. Tehn.-Teor. Lit.,], Moscow-Leningrad, 1950 (Russian). MR**0039790****[9]**W. Rogosinski,*Fourier Series*, Chelsea, New York, 1959.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0330843-6

Keywords:
Function and first derivative,
near minimax norms,
polynomial approximation,
orthogonal basis of polynomials,
generalized Chebyshev polynomial expansions,
position and velocity data,
linear transformation,
almost orthogonal set,
recurrence relation,
normalizing coefficient,
best fit,
integrals of Chebyshev polynomials,
computer programs,
Simpson rule,
Fourier coefficients,
function and derivative deviations,
constraint built into norm,
derivative deviations near endpoints,
convex combination of two integrals,
uniqueness of the approximant,
error bounds and topological properties

Article copyright:
© Copyright 1973
American Mathematical Society