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Shapiro, Harold S.

Regularity properties of the element of closest approximation. (English) Zbl 0273.43005

Trans. Am. Math. Soc. 181, 127-142 (1973).
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Cited in 5 Documents

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
41A50 Best approximation, Chebyshev systems
42A45 Multipliers in one variable harmonic analysis
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References:

[1] V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and F. Riesz, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 1 – 19 (Russian).
[2] James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396 – 414. · Zbl 0015.35604
[3] Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
[4] G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Z. 28 (1928), no. 1, 612 – 634. · JFM 54.0301.03 · doi:10.1007/BF01181186
[5] Lars Hörmander, Estimates for translation invariant operators in \?^{\?} spaces, Acta Math. 104 (1960), 93 – 140. · Zbl 0093.11402 · doi:10.1007/BF02547187
[6] S. Jacobs, An extremal problem for analytic functions in multiply connected regions, Uppsala, Feb. 1969, preprint. (Swedish)
[7] Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). · Zbl 0195.07602
[8] Karel de Leeuw, On \?_{\?} multipliers, Ann. of Math. (2) 81 (1965), 364 – 379. · Zbl 0171.11803 · doi:10.2307/1970621
[9] Eugene Lukacs, Characteristic functions, Griffin’s Statistical Monographs& Courses, No. 5. Hafner Publishing Co., New York, 1960. · Zbl 0087.33605
[10] J. Peetre, Reflections on Besov spaces, Lund, 1966 (mimeographed lecture notes).
[11] Harold S. Shapiro, Applications of normed linear spaces to function-theoretic extremal problems, Lectures on functions of a complex variable, The University of Michigan Press, Ann Arbor, 1955, pp. 399 – 404. · Zbl 0067.30202
[12] Harold S. Shapiro, Smoothing and approximation of functions, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. Revised and expanded edition of mimiographed notes (Matscience Report No. 55); Van Nostrand Reinhold Mathematical Studies.
[13] Harold S. Shapiro, Topics in approximation theory, Springer-Verlag, Berlin-New York, 1971. With appendices by Jan Boman and Torbjörn Hedberg; Lecture Notes in Math., Vol. 187. · Zbl 0213.08501
[14] -, Fourier multipliers whose multiplier norm is an attained value, Proc. Sympos. Linear Operators and Approximation (Oberwolfach, Germany, Aug. 1971) (to appear).
[15] Otto Szász, Über die Fourierschen Reihen gewisser Funktionenklassen, Math. Ann. 100 (1928), no. 1, 530 – 536 (German). · JFM 54.0309.01 · doi:10.1007/BF01448861
[16] Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean \?-space. II. Translation invariant operators, duality, and interpolation, J. Math. Mech. 14 (1965), 821 – 839.
[17] A. Zygmund, On the preservation of classes of functions, J. Math. Mech. 8 (1959), 889-895; erratum 9 (1959), 663. · Zbl 0092.28802
[18] -, Trigonometrical series, 2nd rev. ed., Cambridge Univ. Press, New York, 1968. MR 38 #4882.
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