The convergence of discrete Fourier-Jacobi series

Arenas, Alberto; Ciaurri, Óscar; Labarga, Edgar

doi:10.1090/proc/14925

The convergence of discrete Fourier-Jacobi series
HTML articles powered by AMS MathViewer

by Alberto Arenas, Óscar Ciaurri and Edgar Labarga PDF

Proc. Amer. Math. Soc. 148 (2020), 2539-2550 Request permission

Abstract:

The discrete counterpart of the problem related to the convergence of the Fourier-Jacobi series is studied. To this end, given a sequence, we consider the analogue of the partial sum operator related to Jacobi polynomials and characterize its convergence in the $\ell ^p(\mathbb {N})$-norm.

References

Alberto Arenas, Óscar Ciaurri, and Edgar Labarga, Discrete harmonic analysis associated with Jacobi expansions I: the heat semigroup, preprint, arXiv:1806.00056, 2018.
Alberto Arenas, Óscar Ciaurri, and Edgar Labarga, Discrete harmonic analysis associated with Jacobi expansions II: the Riesz transform, preprint, arXiv:1902.01761, 2019.
Alberto Arenas, Óscar Ciaurri, and Edgar Labarga, A weighted transplantation theorem for Jacobi coefficients, J. Approx. Theory 248 (2019), 105297, 16. MR 4010172, DOI 10.1016/j.jat.2019.105297
Richard Askey, A transplantation theorem for Jacobi coefficients, Pacific J. Math. 21 (1967), 393–404. MR 217509
Richard Askey and Stephen Wainger, A transplantation theorem for ultraspherical coefficients, Pacific J. Math. 16 (1966), 393–405. MR 217508
Jorge J. Betancor, Alejandro J. Castro, Juan C. Fariña, and L. Rodríguez-Mesa, Discrete harmonic analysis associated with ultraspherical expansions, Potential analysis, to appear, https://doi.org/10.1007/s11118-019-09777-9.
S. Chanillo, On the weak behaviour of partial sums of Legendre series, Trans. Amer. Math. Soc. 268 (1981), no. 2, 367–376. MR 632534, DOI 10.1090/S0002-9947-1981-0632534-1
Óscar Ciaurri, T. Alastair Gillespie, Luz Roncal, José L. Torrea, and Juan Luis Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math. 132 (2017), 109–131. MR 3666807, DOI 10.1007/s11854-017-0015-6
José J. Guadalupe, Mario Pérez, and Juan L. Varona, Weak behaviour of Fourier-Jacobi series, J. Approx. Theory 61 (1990), no. 2, 222–238. MR 1050619, DOI 10.1016/0021-9045(90)90005-B
José J. Guadalupe, Mario Pérez, and Juan L. Varona, Weighted weak behaviour of Fourier-Jacobi series, Math. Nachr. 158 (1992), 161–174. MR 1235302, DOI 10.1002/mana.19921580111
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Benjamin Muckenhoupt, Mean convergence of Jacobi series, Proc. Amer. Math. Soc. 23 (1969), 306–310. MR 247360, DOI 10.1090/S0002-9939-1969-0247360-5
Benjamin Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. Math. Soc. 64 (1986), no. 356, iv+86. MR 858466, DOI 10.1090/memo/0356
Jerome Newman and Walter Rudin, Mean convergence of orthogonal series, Proc. Amer. Math. Soc. 3 (1952), 219–222. MR 47811, DOI 10.1090/S0002-9939-1952-0047811-2
F. W. J. Olver (editor-in-chief), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, 2010.
Harry Pollard, The mean convergence of orthogonal series. III, Duke Math. J. 16 (1949), 189–191. MR 28459
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517