Transforms of measures on quotients and spline functions
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- by Alan MacLean PDF
- Trans. Amer. Math. Soc. 261 (1980), 287-296 Request permission
Abstract:
Let G be a LCA group with closed subgroup H and let $v \in M(G/H)$. A general procedure is established for constructing a large family of measures in $M(G)$ whose Fourier transforms interpolate $\hat v$. This method is used to extend a theorem of Shepp and Goldberg by showing that if $v \in M([0, 2\pi ))$, then each even order cardinal spline function which interpolates the sequence $(\hat v(n))_{n = - \infty }^\infty$ Fourier transform of a bounded Borel measure on R.References
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Raouf Doss, On the transform of a singular or an absolutely continuous measure, Proc. Amer. Math. Soc. 19 (1968), 361–363. MR 222569, DOI 10.1090/S0002-9939-1968-0222569-4
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- Richard R. Goldberg, Restrictions of Fourier transforms and extension of Fourier sequences, J. Approximation Theory 3 (1970), 149–155. MR 261269, DOI 10.1016/0021-9045(70)90023-7
- Colin C. Graham and Alan MacLean, A multiplier theorem for continuous measures, Studia Math. 66 (1979/80), no. 3, 213–225. MR 579728, DOI 10.4064/sm-66-3-213-225
- Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae, Quart. Appl. Math. 4 (1946), 45–99. MR 15914, DOI 10.1090/S0033-569X-1946-15914-5
- I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory 2 (1969), 167–206. MR 257616, DOI 10.1016/0021-9045(69)90040-9
- I. J. Schoenberg, Cardinal interpolation and spline functions. II. Interpolation of data of power growth, J. Approximation Theory 6 (1972), 404–420. MR 340899, DOI 10.1016/0021-9045(72)90048-2
- I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0420078
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 261 (1980), 287-296
- MSC: Primary 43A25; Secondary 41A05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0576876-6
- MathSciNet review: 576876