Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
Abstract
In this work we consider some applications of the Weierstrass preparation theorem and Weierstrass pseudopolynomials to study of behavior of the oscillatory integrals and Fourier transforms with analytic and smooth phases with critical points.
First Page
125
Last Page
139
References
1. Arkhipov G.I., Chubarikov V.N., Karatsuba A.A. Trigonometric sums in number theory and analysis. Translated from the 1987 Russian original. de Gruyter Expositions in Mathematics, 39. Walter de Gruyter GmbH Co. KG, Berlin, (2004). – x+554 pp.
2. Arnol'd V.I., Gusein-Zade S.M., Varchenko A.N. Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts. Translated from the Russian by Ian Porteous and Mark Reynolds. Monographs in Mathematics, Vol. 82. Birkhduser Boston, Inc., Boston, MA, (1985).
3. Ikromov I.A. Damped oscillatory integrals and maximal operators. Math. Notes, Vol. 78, Issue 5–6, 773–790 (2005). DOI: 10.1007/s11006-005-0183-z
4. Ikromov I.A., Muranov Sh.A. Estimates of oscillatory integrals with a damping factor. Math. Notes, V. 104, Issue 1–2, 218–230 (2018). DOI: 10.1134/S0001434618070234
5. Littman W. Fourier transform of surface-carried measures and differentiablity of surface averages. Bull. Amer. Math. Soc., Vol. 69, No. 6, 766–770 (1963). DOI: 10.1090/S0002-9904-1963-11025-3
6. Osgood W. Lehrbuch der Funktionentheorie, Bd.II. Teubner, Leipzig, (1929). – 90 pp.
7. Phong D.H., Stein E.M., Sturm J.A. On the growth and stability of real-analytic functions. Amer. J. Math., Vol. 121, No. 3, 519–554 (1999). DOI: 10.1353/ajm.1999.0023
8. Randol B. On the asymptotic behavior of the Fourier transform of the indicator function of a convex set. Trans. Amer. Math. Soc., Vol. 139, 278–285 (1970). DOI: 10.1090/S0002-9947-1969-0251450-5
9. Sadullaev A. Criteria algebraicity of analytic sets. Functional analysis and its application, Vol. 6, Issue 1, 78–79 (1972). DOI: 10.1007/BF01075520
10. Svensson I. Estimates for the Fourier transform of the characteristic function of a convex set. Ark. Mat., Vol. 9, Issue 1–2, 11–22 (1970). DOI: 10.1007/BF02383634
11. Sogge C.D., Stein E.M. Averages of functions over hypersurfaces in ℝn. Invent. Math., Vol. 82, Issue 3, 543–556 (1985). DOI: 10.1007/BF01388869
12. Stein E.M. Harmonic Analysis: real-valued methods, orthogonality and Oscillatory Integrals. Princeton, (1993).
Recommended Citation
Sadullaev, Azimbay and Ikromov, Isroil
(2019)
"Oscillatory integrals and Weierstrass polynomials,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 2:
Iss.
2, Article 4.
DOI: https://doi.org/10.56017/2181-1318.1025