Harmonic analysis
Uniform pointwise estimates for ultraspherical polynomials
Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1239-1250.

We prove pointwise bounds for two-parameter families of Jacobi polynomials. Our bounds imply estimates for a class of functions arising from the spectral analysis of distinguished Laplacians and sub-Laplacians on the unit sphere in arbitrary dimension, and are instrumental in the proof, discussed in a companion paper, of sharp multiplier theorems for those operators.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.255
Classification: 33C45, 33C55, 42C05, 58J50
Casarino, Valentina 1; Ciatti, Paolo 2; Martini, Alessio 3

1 Università degli Studi di Padova, DTG, Stradella san Nicola 3, I-36100 Vicenza, Italy
2 Università degli Studi di Padova, DICEA, Via Marzolo 9, I-35100 Padova, Italy
3 School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
@article{CRMATH_2021__359_10_1239_0,
     author = {Casarino, Valentina and Ciatti, Paolo and Martini, Alessio},
     title = {Uniform pointwise estimates for ultraspherical polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1239--1250},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {10},
     year = {2021},
     doi = {10.5802/crmath.255},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.255/}
}
TY  - JOUR
AU  - Casarino, Valentina
AU  - Ciatti, Paolo
AU  - Martini, Alessio
TI  - Uniform pointwise estimates for ultraspherical polynomials
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1239
EP  - 1250
VL  - 359
IS  - 10
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.255/
DO  - 10.5802/crmath.255
LA  - en
ID  - CRMATH_2021__359_10_1239_0
ER  - 
%0 Journal Article
%A Casarino, Valentina
%A Ciatti, Paolo
%A Martini, Alessio
%T Uniform pointwise estimates for ultraspherical polynomials
%J Comptes Rendus. Mathématique
%D 2021
%P 1239-1250
%V 359
%N 10
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.255/
%R 10.5802/crmath.255
%G en
%F CRMATH_2021__359_10_1239_0
Casarino, Valentina; Ciatti, Paolo; Martini, Alessio. Uniform pointwise estimates for ultraspherical polynomials. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1239-1250. doi : 10.5802/crmath.255. http://www.numdam.org/articles/10.5802/crmath.255/

[1] NIST Digital Library of Mathematical Functions (https://dlmf.nist.gov)

[2] Askey, Richard; Wainger, Stephen Mean convergence of expansions in Laguerre und Hermite series, Am. J. Math., Volume 87 (1965), pp. 695-708 | DOI | Zbl

[3] Axler, Sheldon; Bourdon, Paul; Ramey, Wade Harmonic function theory, Graduate Texts in Mathematics, 137, Springer, 2001 | DOI

[4] Barceló, Juan A.; Ruiz, Alberto; Vega, Luis Weighted estimates for the Helmholtz equation and some applications, J. Funct. Anal., Volume 150 (1997) no. 2, pp. 356-382 | DOI | MR

[5] Boyd, William G. C.; Dunster, T. M. Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions, SIAM J. Math. Anal., Volume 17 (1986), pp. 422-450 | DOI | MR | Zbl

[6] Burq, Nicolas; Dyatlov, Semyon; Ward, Rachel; Zworski, Maciej Weighted eigenfunction estimates with applications to compressed sensing, SIAM J. Math. Anal., Volume 44 (2012) no. 5, pp. 3481-3501 | DOI | MR | Zbl

[7] Casarino, Valentina; Ciatti, Paolo; Martini, Alessio From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere, Adv. Math., Volume 350 (2019), pp. 816-859 | DOI | MR | Zbl

[8] Casarino, Valentina; Ciatti, Paolo; Martini, Alessio Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators (2021) (to appear in Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnab007)

[9] Dall’Ara, Gian Maria; Martini, Alessio A robust approach to sharp multiplier theorems for Grushin operators, Trans. Am. Math. Soc., Volume 373 (2020) no. 11, pp. 7533-7574 | DOI | MR | Zbl

[10] Erdélyi, Arthur; Magnus, W.; Oberhettinger, F.; Tricomi, Francesco G. Higher Transcendental Functions. Vol. II, Robert E. Krieger Publishing Company, 1981

[11] Fefferman, Charles; Phong, Duong H. Subelliptic eigenvalue problems, Harmonic analysis. Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, 1981) (The Wadsworth Mathematics Series), Wadsworth International Group, 1983, pp. 590-606 | Zbl

[12] Frank, Rupert L.; Sabin, Julien Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math., Volume 317 (2017), pp. 157-192 | DOI | MR | Zbl

[13] Haagerup, Uffe; Schlichtkrull, Henrik Inequalities for Jacobi polynomials, Ramanujan J., Volume 33 (2014) no. 2, pp. 227-246 | DOI | MR | Zbl

[14] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Volume 121 (1968), pp. 193-218 | DOI | MR | Zbl

[15] Horwich, Adam D.; Martini, Alessio Almost everywhere convergence of Bochner–Riesz means on Heisenberg-type groups, J. Lond. Math. Soc., Volume 103 (2021) no. 3, pp. 1066-1119 | DOI | MR | Zbl

[16] Koornwinder, Tom; Kostenko, Aleksey; Teschl, Gerald Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator, Adv. Math., Volume 333 (2018), pp. 796-821 | DOI | MR | Zbl

[17] Krasikov, Ilia On the Erdélyi–Magnus–Nevai conjecture for Jacobi polynomials, Constr. Approx., Volume 28 (2008) no. 2, pp. 113-125 | DOI | Zbl

[18] Krasikov, Ilia On approximation of ultraspherical polynomials in the oscillatory region, J. Approx. Theory, Volume 222 (2017), pp. 143-156 | DOI | MR | Zbl

[19] Landau, Lawrence J. Bessel functions: monotonicity and bounds, J. Lond. Math. Soc., Volume 61 (2000) no. 1, pp. 197-215 | DOI | MR | Zbl

[20] Lohöfer, Georg Inequalities for Legendre functions and Gegenbauer functions, J. Approx. Theory, Volume 64 (1991) no. 2, pp. 226-234 | DOI | MR | Zbl

[21] Lohöfer, Georg Inequalities for the associated Legendre functions, J. Approx. Theory, Volume 95 (1998) no. 2, pp. 178-193 | DOI | MR | Zbl

[22] Martini, Alessio; Müller, Detlef; Nicolussi Golo, Sebastiano Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type (2019) (to appear in J. Eur. Math. Soc., https://arxiv.org/abs/1812.02671)

[23] Muldoon, Martin E.; Spigler, Renato Some remarks on zeros of cylinder functions, SIAM J. Math. Anal., Volume 15 (1984), pp. 1231-1233 | DOI | MR | Zbl

[24] Nevai, Paul; Erdélyi, Tamás; Magnus, Alphonse P. Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal., Volume 25 (1994) no. 2, pp. 602-614 | DOI | MR | Zbl

[25] Olver, Frank W. J. Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press Inc., 1974

[26] Olver, Frank W. J. Legendre functions with both parameters large, Philos. Trans. R. Soc. Lond., Ser. A, Volume 278 (1975), pp. 175-185 | MR | Zbl

[27] Olver, Frank W. J. Second order linear differential equations with two turning points, Philos. Trans. R. Soc. Lond., Ser. A, Volume 278 (1975), pp. 137-174 | MR | Zbl

[28] Rauhut, Holger; Ward, Rachel Sparse recovery for spherical harmonic expansions (2011) (SampTA 2011 Conference Proceedings https://arxiv.org/abs/1102.4097)

[29] Seeger, Andreas; Sogge, Christopher D. On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J., Volume 59 (1989) no. 3, pp. 709-736 | MR | Zbl

[30] Sogge, Christopher D. Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 123-138 | DOI | MR | Zbl

[31] Stein, Elias M.; Weiss, Guido Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32, Princeton University Press, 1971

[32] Szegö, Gabor Orthogonal polynomials, Colloquium Publications, 23, American Mathematical Society, 1975 | MR

[33] Thangavelu, Sundaram Lectures on Hermite and Laguerre Expansions, Mathematical Notes, 42, Princeton University Press, 1993 | DOI

[34] Titchmarsh, Edward C. Eigenfunction expansions associated with second-order differential equations. Part I, Clarendon Press, 1962

[35] Vilenkin, Naum Ja. Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, 22, American Mathematical Society, 1968 | DOI

Cited by Sources: