Characterization of functions in terms of rate of convergence of a quadrature process

Authors:
Q. I. Rahman and G. Schmeisser

Journal:
Proc. Amer. Math. Soc. **111** (1991), 85-94

MSC:
Primary 41A25; Secondary 41A55, 46E35, 65D32

MathSciNet review:
1045597

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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the relationship between the regularity properties of a continuous -periodic function and the behavior of the remainder of the trapezoidal rule on applied to .

**[1]**G. R. Blakley, I. Borosh, and C. K. Chui,*A two-dimensional mean problem*, J. Approximation Theory**22**(1978), no. 1, 11–26. MR**0466561****[2]**Itshak Borosh and Charles K. Chui,*On characterization of functions by their Gauss-Chebyshev quadratures*, SIAM J. Math. Anal.**10**(1979), no. 3, 532–541. MR**529071**, 10.1137/0510050**[3]**Paul L. Butzer and Rolf J. Nessel,*Fourier analysis and approximation*, Academic Press, New York-London, 1971. Volume 1: One-dimensional theory; Pure and Applied Mathematics, Vol. 40. MR**0510857****[4]**Chin Hung Ching,*On averages of a function and its application to boundary value problems with insufficient data*, SIAM J. Math. Anal.**5**(1974), 941–947. MR**0385427****[5]**Chin Hung Ching and Charles K. Chui,*Analytic functions characterized by their means on an arc*, Trans. Amer. Math. Soc.**184**(1973), 175–183 (1974). MR**0338375**, 10.1090/S0002-9947-1973-0338375-2**[6]**Chin Hung Ching and Charles K. Chui,*Mean boundary value problems and Riemann series*, J. Approximation Theory**10**(1974), 324–336. MR**0382661****[7]**Charles K. Chui and Philip W. Smith,*Characterization of a function by certain infinite series it generates*, Pacific J. Math.**53**(1974), 363–371. MR**0357791****[8]**Ernst Hellinger and Otto Toeplitz,*Grundlagen für eine Theorie der unendlichen Matrizen*, Math. Ann.**69**(1910), no. 3, 289–330 (German). MR**1511591**, 10.1007/BF01456325**[9]**J. H. Loxton and J. W. Sanders,*The kernel of a rule of approximate integration*, J. Austral. Math. Soc. Ser. B**21**(1979/80), no. 3, 257–267. MR**551429**, 10.1017/S0334270000002356**[10]**Günter Meinardus,*Approximation of functions: Theory and numerical methods*, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR**0217482****[11]**Aurel Wintner,*Diophantine approximations and Hilbert’s space*, Amer. J. Math.**66**(1944), 564–578. MR**0011497****[12]**A. A. Žensykbaev,*Best quadrature formula for the class*, Analysis Mathematica**3**(1977), 83-95.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1045597-0

Keywords:
Quadrature formula,
trapezoidal rule,
speed of convergence,
regularity properties,
function spaces

Article copyright:
© Copyright 1991
American Mathematical Society