Uniform estimates for Mittag--Leffler functions with smooth phase

Akbar R. Safarov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 5, pp. 673-680.

Résumé

In this paper we consider the problem on uniform estimates for Mittag–Leffler functions with the smooth phase functions having singularities $D_{\infty} $, $D_{4}^{\pm}$ and $A_{r}$. The generalisation is that we replace the exponential function with the Mittag–Leffler-type function, to study oscillatory type integrals.

Détail

Keywords: Mittag–Leffler functions, phase function
Mots-clés : amplitude.

Bibliographie

[1] V.I.Arnold, S.M.Gusein-Zade, A.N.Varchenko, Singularities of Differentiable Maps, Birkhauser, Boston–Basel–Stuttgart, 1985 | MR | Zbl

[2] G.I.Arkhipov, A.A.Karatsuba, V.N.Chubarikov, Theory of multiple trigonometric sums, Nauka, M., 1987 | MR | Zbl

[3] A.N.Varchenko, “Newton polyhedra and estimation of oscillating integrals”, Functional Analysis and Its Applications, 10 (1976), 175–196 | DOI | MR

[4] J.Duistermaat, “Oscillatory integrals Lagrange immersions and unifoldings of singularities”, Comm. Pure. Appl. Math., 27:2 (1974), 207–281 | DOI | MR | Zbl

[5] K.G. Van der Korput, “Zur Methode der stationaren phase”, Compositio Math., 1 (1934), 15–38 | MR | Zbl

[6] V.N.Karpushkin, “Uniform estimates for oscillatory integrals with parabolic or hyperbolic phase”, Proceedings of the I.G.Petrovsky Seminar, 9, 1983, 3–39 (in Russian) | MR | Zbl

[7] M.Ruzhansky, B.Torebek, Van der Corput lemmas for Mittag-Leffler functions I. $\alpha-$directions, arXiv: 2002.07492 | MR

[8] I.Podlubny, Fractional Differensial Equations, Academic Press, New York, 1999 | MR

[9] M.G.Mittag-Leffler, “Sur l'intégrale de Laplace-Abel”, C. R. Acad. Sci. Paris, 135 (1902), 937–939

[10] M.G.Mittag-Leffler, “Une généralization de l'intégrale de Laplace-Abel”, Comp. Rend. Acad. Sci. Paris, 136 (1903), 537–539

[11] M.G.Mittag-Leffler, “Sur la nouvelle fonction $E_{\alpha}(x)$”, Comp. Rend. Acad. Sci. Paris, 137 (1903), 554–558

[12] M.G.Mittag-Leffler, “Sopra la funzione $E_{\alpha}(x)$”, Rend. R. Acc. Lincei, 13 (1904), 3–5

[13] P.Humbert, “Quelques résultats relatifs à la fonction de Mittag-Leffler”, C. R. Acad. Sci. Paris, 236 (1953), 1467–1468 | MR | Zbl

[14] R.P.Agarwal, “A propos d'une note de M.Pierre Humbert”, C. R. Acad. Sci. Paris, 236 (1953), 2031–2032 | MR | Zbl

[15] P.Humbert, R.P.Agarwal, “Sur la fonction de Mittag-Leffler et quelquenes de ses génèralisationes”, Bull. Sci. Math., Ser. II, 77 (1953), 180–185 | MR | Zbl

[16] M.M.Dzherbashyan, “On the asymtotic expansion of a function of Mittag-Leffler type”, Akad. Nauk Armjan. SSR Doklady, 19 (1954), 65–72 (in Russian) | MR

[17] M.M.Dzherbashyan, “On integral representation of functions continuous on given rays (generalization of the Fourier integrals)”, Izvestija Akad. Nauk SSSR Ser. Mat., 18 (1954), 427–448 (in Russian) | MR

[18] M.M.Dzherbashyan, “On Abelian summation of the eneralized integral transform”, Akad. Nauk Armjan. SSR Izvestija, fiz-mat. estest. techn.nauki, 7:6 (1954), 1–26 (in Russian) | MR

[19] R.Gorenflo, A.Kilbas, Francesco Mainardi, S.Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin–Heidelberg, 2014 | DOI | MR | Zbl

[20] A.Safarov, “Invariant estimates of two-dimensional oscillatory integrals”, Math. Notes, 104 (2018), 293–302 | DOI | MR | Zbl

[21] A.Safarov, “On invariant estimates for oscillatory integrals with polynomial phase”, J. Sib. Fed. Univ. Math. Phys., 9 (2016), 102–107 | DOI

[22] N.N.Yanenko, “On the weak approximation of the differential equations systems”, Siberian Math. J., 5:6 (1964), 1431–1434 (in Russian) | MR | Zbl

[23] N.N.Yanenko, G.V.Demidov, “The research of a Cauchy problem by method of weak approximation”, Dokl. Akad. Nauk SSSR, 6 (1966), 1242–1244 (in Russian) | MR