2011

On the rate of Lp -convergence of Balakrishnan Rubin-type hypersingular integrals associated to the Gauss-Weierstrass semigroup

We introduce a family of Balakrishnan Rubin-type hypersingular integrals depending on a parameter ε and generated by the Gauss Weierstrass semigroup. Then the connection between the order of Lp smoothness of a Lp function φ and the rate of Lp -convergence of these families to φ, as ε tends to 0, is obtained.

PDF

___

[1] Aliev IA, Rubin B. Wavelet-like transforms for admissible semi-groups; inversion formulas for potentials and Radon transforms. J Fourier Anal Appl 2005; 11: 333-352.
[2] Aliev IA. Bi-parametric potentials, relevant function spaces and wavelet-like transforms. Integr Equat Oper Th 2009; 65: 151-167.
[3] Aliev IA, Rubin B, Uyhan (Bayrakci) S, Sezer S. Composite Wavelet Transforms: Applications and Perspectives, Radon Transforms, Geometry, and Wavelets. Book Series: Contemporary Mathematics, Amer Math Soc 2008; 464: 1-25.
[4] Aliev IA, Eryigit M. Inversion of Bessel potentials with the aid of weighted wavelet transforms. Math Nachr 2002; 242: 27-37.
[5] Aliev IA, Bayrakci S. On inversion of Bessel potentials associated with the Laplace-Bessel differential operator. Acta Math Hungar 2002; 95: 125-145.
[6] Aliev IA, C¸ obano˘glu, S. The rate of convergence of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroups. Integ Transf Spec F 2014; 25: 943-954.
[7] Aliev IA, Eryigit M. On a rate of convergence of truncated hypersingular integrals associated to Riesz and Bessel potentials. J Math Anal Appl 2013; 406: 352-359.
[8] Aliev IA. On the Bochner-Riesz summability and restoration of µ-smooth functions by means of their Fourier transforms. Fractional Calculus and Applied Analysis 1999; 2: 265-277.
[9] Devore RA, Lorentz GG. Constructive Approximation. Berlin, Germany: Springer-Verlag, 1993.
[10] Fisher MJ. Singular integrals and fractional powers of operators. Trans Amer Math Soc 1971; 161: 307-326.
[11] Hajibayov MG. Generalized potentials on commutative hypergroups. Azerbaijan Journal of Mathematics 2015; 5: 37-46.
[12] Kokilashvili VM, Kufner A. Fractional integrals on spaces of homogeneous type. Comment Math Univ Carolinae 1989; 30: 511-523.
[13] Lizorkin PI. Characterization of the spaces L r p (R n ) in terms of difference singular integrals. Mat Sb (N.S.) 1970; 81: 79-91 (in Russian).
[14] Landkof NS. Foundations of Modern Potential Theory. Translated from Russian by A.P. Doohovskoy, Grundlehren Der Mathematischen Wissenschaften in Einzeldarst Band 180, New York, NY, USA: Springer-Verlag, 1972.
[15] Nakai E. On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type. Sci Math Japonicae 2001; 54: 473-487.
[16] Nakai E, Sumitomo H. On generalized Riesz potentials and spaces of some smooth functions. Sci Math Japonicae 2001; 54: 463-472.
[17] Rubin B. A method of characterization and inversion of Bessel and Riesz potentials. Sov Math 1986; 30: 78-89.
[18] Rubin B. Inversion of potentials on R n with the aid of Gauss-Weierstrass integrals. Math Notes 1987; 41: 22-27. English translation from Math Zametki 1987; 41: 34-42.
[19] Rubin B. Fractional integrals and potentials. Harlow, UK: Pitman Monographs and Surveys in Pure and Applied Mathematics, 1996.
[20] Rubin B. Fractional calculus and wavelet transforms in integral geometry. Fract Calc Appl Anal 1998; 1: 193-219.
[21] Samko SG. Spaces L α p,r(R n ) and hypersingular integrals. Studia Math 1977; 61: 193-230.
[22] Samko SG. Hypersingular Integrals and Their Applications. In: Ser.: Analytical Methods and Special Functions. London, UK: Taylor and Francis, 2002.
[23] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. London, UK: Gordon and Breach, Sci. Publ., 1993.
[24] Stein E. Singular Integrals and Differentiability Properties of Functions. Princeton, NJ, USA: Princeton University Press, 1970.
[25] Stein E. The characterization of functions arising as potentials, I. Bull Amer Math Soc 1961; 67: 102-104.
[26] Stein E, Weiss G. Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ, USA: Princeton University Press, 1971.
[27] Sezer S, Aliev IA. A new characterization of the Riesz potentials spaces with the aid of composite wavelet transform. J Math Anal Appl 2010; 372: 549-558.
[28] Sezer S, Aliev IA. On the Gauss Weierstrass summability of multiple trigonometric series at µ-smoothness points. Acta Mathematica Sinica, English series 2011, 27: 741-746.
[29] Wheeden RL. On hypersingular integrals and Lebesgue spaces of differentiable functions. Trans Amer Math Soc 1968; 134: 421-435.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv