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Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Volume: 50 Issue: 6

On the approximation properties of bi-parametric potential-type integral operators

Çağla Sekin(AKDENIZ UNIVERSITY)0000-0001-7176-5164
Mutlu Güloğlu(AKDENIZ UNIVERSITY)0000-0003-4197-1287
İlham Aliyev(AKDENIZ UNIVERSITY)0000-0003-2353-7700
Pages
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Publication Date
2021-12-14
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⚡ TL;DR(Too Long; Didn't Read)

Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=O(1)(\alpha -\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds when $A^{\alpha}=J_{\beta }^{\alpha }$.

📌 3 Key Takeaways

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Objective & Hypothesis:In this work we study the approximation properties of the classical Riesz potentials $I^{\alpha }f\equiv (-\Delta )^{-\alpha /2}f$ and the so-called bi-parametric potential-type operators $J_{\beta }^{\alpha }f\equiv(E+(-\Delta )^{\beta /2})^{-\alpha /\beta }f$ as $\alpha \rightarrow \alpha_{0}>0$ where, $\alpha >0$, $\beta >0$, $E$ is the identity operator and $\Delta $ is the laplacian.
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Key Findings & Data:These potential-type operators generalize the famous Bessel potentials when $\beta =2$ and Flett potentials when $\beta =1$.
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Conclusion & Impact:Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=O(1)(\alpha -\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds when $A^{\alpha}=J_{\beta }^{\alpha }$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Abel-Poisson semigroup · Gauss-Weierstrass semigroup · Riesz potentials

Abstract

In this work we study the approximation properties of the classical Riesz potentials $I^{\alpha }f\equiv (-\Delta )^{-\alpha /2}f$ and the so-called bi-parametric potential-type operators $J_{\beta }^{\alpha }f\equiv(E+(-\Delta )^{\beta /2})^{-\alpha /\beta }f$ as $\alpha \rightarrow \alpha_{0}>0$ where, $\alpha >0$, $\beta >0$, $E$ is the identity operator and $\Delta $ is the laplacian. These potential-type operators generalize the famous Bessel potentials when $\beta =2$ and Flett potentials when $\beta =1$. We show that, if $A^{\alpha}$ is one of operators $J_{\beta }^{\alpha }$ or $I^{\alpha}$, then at every Lebesgue point of $f\in L_{p}(\mathbb{R}^{n})$ the asymptotic equality $(A^{\alpha}f)(x)-(A^{\alpha _{0}}f)(x)=O(1)(\alpha-\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds. Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=O(1)(\alpha -\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds when $A^{\alpha}=J_{\beta }^{\alpha }$.

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Cite This Article

Sekin, Ç., Güloğlu, M., Aliyev, İ. (2021). On the approximation properties of bi-parametric potential-type integral operators. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 6. https://doi.org/10.15672/hujms.821159

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References (18)

  1. İ.A. Aliev, Bi-parametric potentials, relevant function spaces and wavelet-like trans- forms, Integr. Equ. Oper. Theory 65, 151-167, 2009.
  2. İ.A. Aliev, A.D. Gadjiev and A. Aral, On approximation properties of a family of linear operators at critical value of parameter, J. Approx. Theory 138, 242-253, 2006.
  3. İ.A. Aliev and B. Rubin, Wavelet-like transforms for admissible semi-groups; inver- sion formulas for potentials and Radon transforms, J. Fourier Anal. Appl. 11 (3), 333-352, 2005.
  4. İ.A. Aliev, B. Rubin, S. Sezer and S. Uyhan, Composite wavelet transforms: applica- tions and perspectives, Contemp. Math., AMS 464, 1-27, 2008.
  5. İ.A. Aliev, S. Sezer and M. Eryiğit, An integral transform associated to the Poisson integral and inversion of Flett potentials, J. Math. Anal. Appl. 321 (2), 691-704, 2006.
  6. N. Aronszajn and K.T. Smith, Theory of Bessel Potentials, I, Ann. Inst. Fourier 11, 385-475, 1961.
  7. T.M. Flett, Temperatures, Bessel Potentials and Lipschitz spaces, Proc. London. Math. Soc. 22, 385-451, 1971.
  8. A.D. Gadjiev, A. Aral and İ.A. Aliev, On behaviour of the Riesz and generalized Riesz potentials as order tends to zero, Math. Inequal. Appl. 10 (4), 875-888, 2007.
  9. S.G. Gal, Approximation by complex potentials generated by the Gamma function, Turkish J. Math. 35, 443-456, 2011.
  10. R. Johnson, Temperatures, Riesz potentials and the Lipschitz spaces of Herz, Proc. London Math. Soc. 3 (27), 290-316, 1973.
  11. T. Kurokawa, On the Riesz and Bessel kernels as approximations of the identity, Sci. Rep. Kagoshima Univ. 30, 31-45, 1981.
  12. B. Rubin, Fractional Integrals and Potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics 82, Longman, Harlow, 1996.
  13. S.M. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Sci. Publ. London-New York, 1993.
  14. S. Sezer, On approximation properties of the families of Flett and generalized Flett potentials, Int. J. Math. Anal. 3 (39), 1905-1915, 2009.
  15. S. Sezer and İ.A. Aliev, A new characterization of the Riesz potential spaces with aid of a composite wavelet transform, J. Math. Anal. Appl. 372, 549-558, 2010.
  16. E. Stein, The characterization of function arising as potentials, I, Bull. Amer. Math. Soc. 67 (1), 101-104, 1961.
  17. E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970.
  18. S. Uyhan, A.D. Gadjiev and İ.A. Aliev, On approximation properties of the parabolic potentials, Bull. Aust. Math. Soc. 74 (3), 449-460, 2006.