Multiplier criteria of Marcinkiewicz type for Jacobi expansions
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- by George Gasper and Walter Trebels PDF
- Trans. Amer. Math. Soc. 231 (1977), 117-132 Request permission
Abstract:
It is shown how an integral representation for the product of Jacobi polynomials can be used to derive a certain integral Lipschitz type condition for the Cesàro kernel for Jacobi expansions. This result is then used to give criteria of Marcinkiewicz type for a sequence to be multiplier of type (p, p), $1 < p < \infty$, for Jacobi expansions.References
-
W. N. Bailey, The generating function of Jacobi polynomials, J. London Math. Soc. 13 (1938), 8-12.
- H. Bavinck and W. Trebels, On $M_{p}^{q}$ multipliers for Jacobi expansions, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976) Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 101–112. MR 540293
- Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223–263 (French). MR 338697, DOI 10.1090/S0002-9947-1973-0338697-5
- L. S. Bosanquet, Note on the Bohr-Hardy theorem, J. London Math. Soc. 17 (1942), 166–173. MR 7800, DOI 10.1112/jlms/s1-17.3.166
- Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
- William C. Connett and Alan L. Schwartz, A multiplier theorem for ultraspherical series, Studia Math. 51 (1974), 51–70. MR 358209, DOI 10.4064/sm-51-1-51-70
- William C. Connett and Alan L. Schwartz, A multiplier theorem for Jacobi expansions, Studia Math. 52 (1974/75), 243–261. MR 387934, DOI 10.4064/sm-52-3-243-261 A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, vol. I, McGraw-Hill, New York, 1953. MR 15, 419.
- George Gasper, Positivity and the convolution structure for Jacobi series, Ann. of Math. (2) 93 (1971), 112–118. MR 284628, DOI 10.2307/1970755
- George Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), 261–280. MR 310536, DOI 10.2307/1970800
- George Gasper, Positive sums of the classical orthogonal polynomials, SIAM J. Math. Anal. 8 (1977), no. 3, 423–447. MR 432946, DOI 10.1137/0508032
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- Tom Koornwinder, Jacobi polynomials. II. An analytic proof of the product formula, SIAM J. Math. Anal. 5 (1974), 125–137. MR 385198, DOI 10.1137/0505014 J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8 (1939), 78-91.
- B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92. MR 199636, DOI 10.1090/S0002-9947-1965-0199636-9 G. Szegö, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967. (1st ed., 1939; MR 1, 14)
- Walter Trebels, Multipliers for $(C, \alpha )$-bounded Fourier expansions in Banach spaces and approximation theory, Lecture Notes in Mathematics, Vol. 329, Springer-Verlag, Berlin-New York, 1973. MR 0510852
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 231 (1977), 117-132
- MSC: Primary 42A18
- DOI: https://doi.org/10.1090/S0002-9947-1977-0467139-8
- MathSciNet review: 0467139