Summability of Hermite expansions. I
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Abstract:
We study the summability of one-dimensional Hermite expansions. We prove that the critical index for the Riesz summability is $1/6$. We also prove analogues of the Fejér-Lebesgue theorem and Riemann’s localisation principle.References
- Richard Askey, Norm inequalities for some orthogonal series, Bull. Amer. Math. Soc. 72 (1966), 808–823. MR 198114, DOI 10.1090/S0002-9904-1966-11568-9
- Richard Askey and I. I. Hirschman Jr., Mean summability for ultraspherical polynomials, Math. Scand. 12 (1963), 167–177. MR 164198, DOI 10.7146/math.scand.a-10680
- Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695–708. MR 182834, DOI 10.2307/2373069
- Robert Campbell, Détermination effective de toutes les moyennes de Cesàro d’ordre entier pour des séries de polynomes orthogonaux comprenant ceux de Laguerre et de Hermite, C. R. Acad. Sci. Paris 243 (1956), 882–885 (French). MR 79669 A. Erdelyi et al., Asymptotic forms for Laguerre polynomials, Golden Jubilee volume of the Indian Math. Soc.
- Dieter Ernst, Über die Konvergenz der $(C,\,1)$-Mittel von Fourier-Laguerre-Reihen, Compositio Math. 21 (1969), 81–101 (German, with English summary). MR 241894
- G. Freud, On weighted simultaneous polynomial approximation, Studia Sci. Math. Hungar. 7 (1972), 337–342. MR 330844
- Géza Freud, On polynomial approximation with respect to general weights, Functional analysis and its applications (Internat. Conf., Eleventh Anniversary of Matscience, Madras, 1973; dedicated to Alladi Ramakrishnan), Lecture Notes in Math., Vol. 399, Springer, Berlin, 1974, pp. 149–179. MR 0404924
- G. Freud and S. Knapowski, On linear processes of approximation. III, Studia Math. 25 (1965), 373–383. MR 181863, DOI 10.4064/sm-25-3-373-384
- J. J. Gergen, Summability of double Fourier series, Duke Math. J. 3 (1937), no. 2, 133–148. MR 1545976, DOI 10.1215/S0012-7094-37-00310-7
- Lars Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966) Yeshiva Univ., Belfer Graduate School of Science, New York, 1969, pp. 155–202. MR 0257589 E. Kogbetliantz, Sur les moyennes arithmetiques des series-noyaux des developpement en series d’Hermite et de Laguerre et sur celles de les series-noyaux drivees term a terme, J. Math. Phys. 14 (1935), 37-99.
- Lee Lorch, The Lebesgue constants for Jacobi series. I, Proc. Amer. Math. Soc. 10 (1959), 756–761. MR 109981, DOI 10.1090/S0002-9939-1959-0109981-3
- C. Markett, Norm estimates for Cesàro means of Laguerre expansions, Approximation and function spaces (Gdańsk, 1979) North-Holland, Amsterdam-New York, 1981, pp. 419–435. MR 649449
- C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1982), no. 1, 19–37 (English, with Russian summary). MR 662701, DOI 10.1007/BF02073769
- C. Markett, Norm estimates for $(C,\,\delta )$ means of Hermite expansions and bounds for $\delta _{\textrm {eff}}$, Acta Math. Hungar. 43 (1984), no. 3-4, 187–198. MR 733853, DOI 10.1007/BF01958018
- Benjamin Muckenhoupt, Mean convergence of Hermite and Laguerre series. I, II, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid. 147 (1970), 433–460. MR 0256051, DOI 10.1090/S0002-9947-1970-0256051-9
- Benjamin Muckenhoupt, Mean convergence of Hermite and Laguerre series. I, II, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid. 147 (1970), 433–460. MR 0256051, DOI 10.1090/S0002-9947-1970-0256051-9
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
- J. Peetre, Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand. 15 (1964), 83–92. MR 178252, DOI 10.7146/math.scand.a-10731
- Eileen L. Poiani, Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1–31. MR 310537, DOI 10.1090/S0002-9947-1972-0310537-9
- Harry Pollard, The mean convergence of orthogonal series. I, Trans. Amer. Math. Soc. 62 (1947), 387–403. MR 22932, DOI 10.1090/S0002-9947-1947-0022932-1
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 —, Beijing lectures on Fourier analysis, Ann. of Math. Studies, no. 112, Princeton Univ. Press, Princeton, N.J., 1986.
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Gábor Szegő, Orthogonal polynomials, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0310533
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 314 (1989), 119-142
- MSC: Primary 42C10; Secondary 42A24
- DOI: https://doi.org/10.1090/S0002-9947-1989-99923-2
- MathSciNet review: 958904