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On estimates for trigonometric integrals with quadratic phase

https://doi.org/10.22405/2226-8383-2024-25-1-52-61

Abstract

This paper is devoted to the summation problem for trigonometric integrals with quadratic phase. The particular cases of this problem were considered in [2],[3],[4]. We generalize the results of these papers to the multidimensional exponential integrals.

About the Authors

Isroil Akramovich Ikromov
V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan (Tashkent, Uzbekistan), Samarkand State University (Samarkand, Uzbekistan).
Uzbekistan


Akbar Rakhmanovich Safarov
V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan (Tashkent, Uzbekistan), Samarkand State University (Samarkand, Uzbekistan).
Uzbekistan


Akmal Tolliboevich Absalamov
Samarkand State University (Samarkand, Uzbekistan).
Russian Federation


References

1. Arkhipov, G. I., Karatsuba, A.A.& Chubarikov, V.N., 1987. “Theory of multiple trigonometric sums”, Moscow. Nauka, p. 357.

2. Arkhipov, L. G., & Chubarikov, V.N, 2019. “Chebyshevskiy sbornik”, On the exponents of the convergence of singular integrals and singular series of a multivariate problem, vol. 20, no. 4, pp.46–57.

3. Chahkiev, M. A., 2019. “Estimation of the convergence index of a singular integral Terry problems for a homogeneous polynomial degree n of two variables”, LXI International Scientific Readings (in memory of A.N.Kolmogorov) International Scientific and Practical Conference December 16, pp.18–21.

4. Jabbarov, I. Sh., 2019. “Mathematical Notes”, Exponent of a special integral in the twodimensional Tarry problem with homogeneous of degree 2, vol. 105, no. 3, pp. 375—382.

5. Hua Loo-keng, 1952. “On the number of solutions of Tarry’s problem”, Acta Sci. Sinica, vol.1, No. 1, pp. 1–76.

6. Ikromov, I. A., 1997. “On the convergence exponent of trigonometric integrals”, Proceedings, MIRAN, vol.218, pp.179–189.

7. Safarov, A., 2019. “On the 𝐿𝑝-bound for trigonometric integrals”, Analysis mathematica no. 45, pp. 153–176.

8. Safarov, A., 2015. “About summation of oscillatory integrals with homogeneous polynomial of third degree”, Uzbek Mathematical journal no.4 , pp.108–117.

9. Safarov, A., 2018. “Invariant estimates of two-dimensional oscillatory integrals”, Math. Notes, 104, pp.293–302.

10. Safarov, A., 2016. “On invariant estimates for oscillatory integrals with polynomial phase”, J. Sib. Fed. Univ. Math. Phys., 9, pp.102–107.

11. Safarov, A., 2019. “On a problem of restriction of Fourier transform on a hypersurface”, Russian Mathematics, 63 (4), pp.57–63.

12. Safarov, A. R., 2022. “Estimates for Mittag-–Leffler Functions with Smooth Phase Depending on Two Variables”, J. Sib. Fed. Univ. Math. Phys., 15(4), pp.459-–466.

13. Makenhaupt, G., 1996. “Bounds in Lebesgue Spaces of Oscillatory Integral Operators”, Habilitationsschift zur Erlangung der Lehrbefugnis im Fach Matematik der Gesamthochschule, Siegen.

14. Stein, E. M., 1993. “Harmonic Analysis: real-valued methods, orthogonality and Oscillatory Integrals”, Princeton.

15. Vinogradov, I. M., 1980. “Method trigonometric sums in number theory”, Moscow, Nauka, pp. 158.

16. Jong-Guk Bak, Sanghyuk Lee, 2004. “Restriction of the Fourier transform to a quadratic surface in R_𝑛”, Mathematische Zeitschrift № 247, pp.409–422.

17. Lebedev, V. V., 2013. “On the Fourier transform of the characteristic functions of domains with 𝐶^1 boundary”, Func. anal. and its appl. Vol. 47, no. 1. pp. 33–46.

18. Lebedev, V. V., 2013. “Superposition operators in some spaces of the harmonic analyzer the translator”, Dissertation to take The dissertation on competition of a scientific degree of physical and mathematical sciences. URL: https://www.dissercat.com/content/operatory-superpozitsiiv-nekotorykh-prostranstvakh-garmonicheskogo-analiza


Review

For citations:


Ikromov I.A., Safarov A.R., Absalamov A.T. On estimates for trigonometric integrals with quadratic phase. Chebyshevskii Sbornik. 2024;25(1):52-61. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-1-52-61

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