Abstract
In this paper, we prove a multiple analog of the theorem proved by Arkhipov and the author in 1987, which provides an estimate for the discrete Hilbert transform with polynomial phase. For the linear case, the corresponding estimates of the sum of multiple trigonometric series was proved by Telyakovskii.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. A. Telyakovskii, “On estimates of derivatives of trigonometric polynomials of several variables,” Sibirsk. Mat. Zh. [Siberian Math. J.], 4 (1963), no. 6, 1404–1411.
S. A. Telyakovskii, “The uniform boundedness of trigonometric polynomials of several variables,” Mat. Zametki [Math. Notes], 42 (1987), no. 1, 33–39.
S. A. Telyakovskii and V. N. Temlyakov, “On the convergence of Fourier series of bounded variation in several variables,” Mat. Zametki [Math. Notes], 61 (1997), no. 4, 583–591.
G. I. Arkhipov and K. I. Oskolkov, “On a special trigonometric series and its applications,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 134(176) (1987), no. 2(10), 147–157.
I. M. Vinogradov, Method of Trigonometric Sums in Number Theory [in Russian], Nauka, Moscow, 1971.
E. Stein and S. Wainger, “Discrete analogues of singular Radon transforms,” Bull. Amer. Math. Soc., 23 (1990), 537–544.
K. I. Oskolkov, “The class of Vinogradov's series and its applications in harmonic analysis,” in: Progress in Approximation Theory, An International Prospective Proceedings of the International Conference on Approximation Theory held on March 19–22, 1990, Springer-Verlag, 1992, pp. 353–402.
K. I. Oskolkov, “Schrödinger equation and oscillatory Hilbert transforms of second degree,” J. Fourier Anal. Appl., 4 (1998), 341–356.
K. I. Oskolkov, “Vinogradov series in the Cauchy problem for the Schrödinger equation,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 200 (1991), 265–288.
A. Zygmund, Trigonometric Series, Second Edition, Cambridge University Press, Cambridge, 1959.
M. Z. Garaev, “On a multiple trigonometric series,” Acta Arithm., 102 (2002), no. 2, 183–187.
E. Stein and S. Wainger, “The estimation of an integral arising in multiplier transformations,” Studia Math., 35 (1970), 101–104.
E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.
H. Weyl, “ Ñber die Gleichverteilung von Zahlen mod Eins,” Math. Ann., 77 (1916), 313–352.
Chen. Jing-run, “On Professor Hua's estimate of exponential sum.,” Sci. Sinica, 20 (1977), 711–719.
S. B. Stechkin, “An estimate of a complete rational trigonometric sum,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 143 (1977), 188–207.
G. I. Arkhipov, “On the Hilbert–Kamke problem,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 48 (1984), no. 1, 3–52.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Oskolkov, K.I. On a Result of Telyakovskii and Multiple Hilbert Transforms with Polynomial Phases. Mathematical Notes 74, 232–244 (2003). https://doi.org/10.1023/A:1025008308955
Issue Date:
DOI: https://doi.org/10.1023/A:1025008308955