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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Interpolation on uniform meshes by the translates of one function and related attenuation factors
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by F. Locher PDF
Math. Comp. 37 (1981), 403-416 Request permission


The exact Fourier coefficients ${c_j}({P_n}f)$ are proportional to the discrete Fourier coefficients $d_j^{(n)}(f)$ if ${P_n}$ is a translation invariant operator which depends only on the values of f on an equidistant mesh of width $2\pi /n$. The proportionality factors which depend only on ${P_n}$ but not on f are called attenuation factors and have been calculated for several operators ${P_n}$ of spline type. Here we analyze first the interpolation problem which is produced by the functions $\sigma ( \bullet - 2\pi j/n),j = 0, \ldots ,n - 1$, where $\sigma$ is a suitable $2\pi$-periodic generating function. It is essential that the associated interpolation matrix is of discrete convolution type. Thus, we can derive conditions guaranteeing the unique solvability of the interpolation problem and representations of the interpolating function. Then the attenuation factors may be expressed in terms of the Fourier coefficients of $\sigma$. We point especially to the case where $\sigma$ is a reproducing kernel in a suitable Hilbert space. Here we get attenuation factors of a new type which are generated by interpolation with analytic functions.
  • Stefan Bergman, The kernel function and conformal mapping, Second, revised edition, Mathematical Surveys, No. V, American Mathematical Society, Providence, R.I., 1970. MR 0507701
  • L. Collatz & W. Quade, "Zur Interpolationstheorie der reellen periodischen Funktionen," S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl., v. 30, 1938, pp. 383-429.
  • Hartmut Ehlich, Untersuchungen zur numerischen Fourieranalyse, Math. Z. 91 (1966), 380–420 (German). MR 207241, DOI 10.1007/BF01110651
  • Walter Gautschi, Attenuation factors in practical Fourier analysis, Numer. Math. 18 (1971/72), 373–400. MR 305641, DOI 10.1007/BF01406676
  • Michael Golomb, Approximation by periodic spline interpolants on uniform meshes, J. Approximation Theory 1 (1968), 26–65. MR 233121, DOI 10.1016/0021-9045(68)90055-5
  • Michael Golomb, Interpolation operators as optimal recovery schemes for classes of analytic functions, Optimal estimation in approximation theory (Proc. Internat. Sympos., Freudenstadt, 1976) Plenum, New York, 1977, pp. 93–138. MR 0481730
  • Peter Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev. 21 (1979), no. 4, 481–527. MR 545882, DOI 10.1137/1021093
  • W. Knauff and R. Kreß, Optimale Approximation linearer Funktionale auf periodischen Funktionen, Numer. Math. 22 (1973/74), 187–205 (German, with English summary). MR 350286, DOI 10.1007/BF01436967
  • Rainer Kreß, Zur numerischen Integration periodischer Funktionen nach der Rechteckregel, Numer. Math. 20 (1972/73), 87–92 (German, with English summary). MR 317524, DOI 10.1007/BF01436645
  • Herbert Meschkowski, Reihenentwicklungen in der mathematischen Physik, Bibliographisches Institut, Mannheim, 1963 (German). MR 0158192
  • Arnold Schönhage, Approximationstheorie, Walter de Gruyter & Co., Berlin-New York, 1971 (German). MR 0277960
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Math. Comp. 37 (1981), 403-416
  • MSC: Primary 65D10; Secondary 42A15
  • DOI:
  • MathSciNet review: 628704