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Fractional order mixed difference operator and its applications in angular approximation

Year 2020, Volume: 49 Issue: 5, 1594 - 1610, 06.10.2020
https://doi.org/10.15672/hujms.569410

Abstract

Lebesgue spaces are considered with Muckenhoupt weights. Fractional order mixed difference operator is investigated to obtain mixed fractional modulus of smoothness in these spaces. Using this modulus of smoothness we give the proof of direct and inverse estimates of angular trigonometric approximation. Also we obtain an equivalence between fractional mixed modulus of smoothness and fractional mixed K-functional.


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Supporting Institution

Balikesir University Research Projects

Project Number

2019/01

References

  • [1] R. Akgün, Polynomial approximation in weighted Lebesgue spaces, East J. Approx. 17(3), 253–266, 2011.
  • [2] R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz spaces, J. Funct. Spaces Appl. 2012, 1–41, ID 982360, 2012.
  • [3] R. Akgün, Realization and characterization of modulus of smoothness in weighted Lebesgue spaces, St. Petersburg Math. J. 26 (5), 741–756, 2015.
  • [4] R. Akgün, Mixed modulus of continuity in Lebesgue spaces with Muckenhoupt weights, Turkish J. Math. 40 (6), 1169–1192, 2016.
  • [5] R. Akgün, Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle, Complex Var. Elliptic Equ. 64 (2), 330–351, 2019.
  • [6] R. Akgün, Nikol’ski, Jackson and Ul’yanov type inequalities with Muckenhoupt weights, arXiv:1709.02928 [math.CA].
  • [7] P.L. Butzer, H. Dyckhoff, E. Görlich, and R.L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes, Canad. J. Math. 29 (4), 781–793, 1977.
  • [8] C. Cottin, Mixed K-functionals: a measure of smoothness for blending-type approximation, Math. Z. 204 (1), 69–83, 1990.
  • [9] E. DiBenedetto, Real Analysis, Second Ed., Birkhäuser, Boston, 2016.
  • [10] A. Guven and D.M. Israfilov, Improved inverse theorems in weighted Lebesgue and Smirnov spaces, Bull. Belg. Math. Soc. Simon Stevin 14 (4), 681–692, 2007.
  • [11] A. Guven and V. Kokilashvili, On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting, J. Inequal. Appl. 2006, Art. 41837, 2006.
  • [12] D.M. Israfilov, Approximation by p-Faber polynomials in the weighted Smirnov class $E^{p}(G,\omega )$ and the Bieberbach polynomials, Constr. Approx. 17 (3), 335–351, 2001.
  • [13] S.Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 33, 95–100, 2010.
  • [14] V. Kokilashvili and Y.E. Yildirir, On the approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 143, 103–113, 2007.
  • [15] A. D. Nakhman and B. P. Osilenker, Estimates of weighted norms of some operators generated by multiple trigonometric Fourier series. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 26 (4), 39–50, 1982.
  • [16] M.K. Potapov, Approximation by "angle” (in Russian). In: Proceedings of the Conference on the Constructive Theory of Functions and Approximation Theory, Budapest, 1969, Akadémiai Kiadó, pp. 371–399, 1972.
  • [17] M.K. Potapov, The Hardy-Littlewood and Marcinkiewicz-Littlewood-Paley theorems, approximation "by an angle”, and the imbedding of certain classes of functions (in Russian), Mathematica (Cluj) 14 (37), 339–362, 1972.
  • [18] M.K. Potapov, A certain imbedding theorem (in Russian), Mathematica (Cluj) 14(37), 123–146, 1972.
  • [19] M.K. Potapov, Approximation "by angle”, and imbedding theorems (in Russian), Math. Balkanica 2, 183–198, 1972.
  • [20] M.K. Potapov, Imbedding of classes of functions with a dominating mixed modulus of smoothness (in Russian), Trudy Mat. Inst. Steklov. 131, 199–210, 1974.
  • [21] M.K. Potapov and B.V. Simonov, On the relations between generalized classes of Besov-Nikolski˘ı and Weyl-Nikolski˘ı functions, Proc. Steklov Inst. Math. 214 (3), 243– 259, 1996.
  • [22] M.K. Potapov, B.V. Simonov, and B. Lakovich, On estimates for the mixed modulus of continuity of a function with a transformed Fourier series, Publ. Inst. Math. (Beograd) (N.S.) 58 (72), 167–192, 1995.
  • [23] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Embedding theorems for Besov- Nikolski˘ı and Weyl-Nikolski˘ı classes in a mixed metric, Moscow Univ. Math. Bull. 59, 19–26, 2005.
  • [24] M.K. Potapov, B.V. Simonov and S.Y. Tikhonov, Transformation of Fourier series using power and weakly oscillating sequences, Math. Notes 77 (1-2), 90–107, 2005.
  • [25] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Relations between mixed moduli of smoothness and embedding theorems for the Nikolski˘ı classes, Proc. Steklov Inst. Math. 269 (1), 197–207, 2010.
  • [26] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Mixed moduli of smoothness in $L_{p},1<p<\infty $: A survey, Surv. Approx. Theory 8, 1–57, 2013.
  • [27] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Translated from Russian 1987, Gordon and Breach Science Publishers, Yverdon, 1993.
  • [28] R. Taberski, Differences, moduli and derivatives of fractional orders, Comment. Math. Prace Mat. 19 (2), 389–400, 1976/77.
  • [29] Y.E. Yildirir and D.M. Israfilov, Approximation theorems in weighted Lorentz spaces, Carpathian J. Math. 26 (1), 108–119, 2010.
Year 2020, Volume: 49 Issue: 5, 1594 - 1610, 06.10.2020
https://doi.org/10.15672/hujms.569410

Abstract

Project Number

2019/01

References

  • [1] R. Akgün, Polynomial approximation in weighted Lebesgue spaces, East J. Approx. 17(3), 253–266, 2011.
  • [2] R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz spaces, J. Funct. Spaces Appl. 2012, 1–41, ID 982360, 2012.
  • [3] R. Akgün, Realization and characterization of modulus of smoothness in weighted Lebesgue spaces, St. Petersburg Math. J. 26 (5), 741–756, 2015.
  • [4] R. Akgün, Mixed modulus of continuity in Lebesgue spaces with Muckenhoupt weights, Turkish J. Math. 40 (6), 1169–1192, 2016.
  • [5] R. Akgün, Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle, Complex Var. Elliptic Equ. 64 (2), 330–351, 2019.
  • [6] R. Akgün, Nikol’ski, Jackson and Ul’yanov type inequalities with Muckenhoupt weights, arXiv:1709.02928 [math.CA].
  • [7] P.L. Butzer, H. Dyckhoff, E. Görlich, and R.L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes, Canad. J. Math. 29 (4), 781–793, 1977.
  • [8] C. Cottin, Mixed K-functionals: a measure of smoothness for blending-type approximation, Math. Z. 204 (1), 69–83, 1990.
  • [9] E. DiBenedetto, Real Analysis, Second Ed., Birkhäuser, Boston, 2016.
  • [10] A. Guven and D.M. Israfilov, Improved inverse theorems in weighted Lebesgue and Smirnov spaces, Bull. Belg. Math. Soc. Simon Stevin 14 (4), 681–692, 2007.
  • [11] A. Guven and V. Kokilashvili, On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting, J. Inequal. Appl. 2006, Art. 41837, 2006.
  • [12] D.M. Israfilov, Approximation by p-Faber polynomials in the weighted Smirnov class $E^{p}(G,\omega )$ and the Bieberbach polynomials, Constr. Approx. 17 (3), 335–351, 2001.
  • [13] S.Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 33, 95–100, 2010.
  • [14] V. Kokilashvili and Y.E. Yildirir, On the approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 143, 103–113, 2007.
  • [15] A. D. Nakhman and B. P. Osilenker, Estimates of weighted norms of some operators generated by multiple trigonometric Fourier series. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 26 (4), 39–50, 1982.
  • [16] M.K. Potapov, Approximation by "angle” (in Russian). In: Proceedings of the Conference on the Constructive Theory of Functions and Approximation Theory, Budapest, 1969, Akadémiai Kiadó, pp. 371–399, 1972.
  • [17] M.K. Potapov, The Hardy-Littlewood and Marcinkiewicz-Littlewood-Paley theorems, approximation "by an angle”, and the imbedding of certain classes of functions (in Russian), Mathematica (Cluj) 14 (37), 339–362, 1972.
  • [18] M.K. Potapov, A certain imbedding theorem (in Russian), Mathematica (Cluj) 14(37), 123–146, 1972.
  • [19] M.K. Potapov, Approximation "by angle”, and imbedding theorems (in Russian), Math. Balkanica 2, 183–198, 1972.
  • [20] M.K. Potapov, Imbedding of classes of functions with a dominating mixed modulus of smoothness (in Russian), Trudy Mat. Inst. Steklov. 131, 199–210, 1974.
  • [21] M.K. Potapov and B.V. Simonov, On the relations between generalized classes of Besov-Nikolski˘ı and Weyl-Nikolski˘ı functions, Proc. Steklov Inst. Math. 214 (3), 243– 259, 1996.
  • [22] M.K. Potapov, B.V. Simonov, and B. Lakovich, On estimates for the mixed modulus of continuity of a function with a transformed Fourier series, Publ. Inst. Math. (Beograd) (N.S.) 58 (72), 167–192, 1995.
  • [23] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Embedding theorems for Besov- Nikolski˘ı and Weyl-Nikolski˘ı classes in a mixed metric, Moscow Univ. Math. Bull. 59, 19–26, 2005.
  • [24] M.K. Potapov, B.V. Simonov and S.Y. Tikhonov, Transformation of Fourier series using power and weakly oscillating sequences, Math. Notes 77 (1-2), 90–107, 2005.
  • [25] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Relations between mixed moduli of smoothness and embedding theorems for the Nikolski˘ı classes, Proc. Steklov Inst. Math. 269 (1), 197–207, 2010.
  • [26] M.K. Potapov, B.V. Simonov, and S.Y. Tikhonov, Mixed moduli of smoothness in $L_{p},1<p<\infty $: A survey, Surv. Approx. Theory 8, 1–57, 2013.
  • [27] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Translated from Russian 1987, Gordon and Breach Science Publishers, Yverdon, 1993.
  • [28] R. Taberski, Differences, moduli and derivatives of fractional orders, Comment. Math. Prace Mat. 19 (2), 389–400, 1976/77.
  • [29] Y.E. Yildirir and D.M. Israfilov, Approximation theorems in weighted Lorentz spaces, Carpathian J. Math. 26 (1), 108–119, 2010.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ramazan Akgün 0000-0001-6247-8518

Project Number 2019/01
Publication Date October 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 5

Cite

APA Akgün, R. (2020). Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics, 49(5), 1594-1610. https://doi.org/10.15672/hujms.569410
AMA Akgün R. Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics. October 2020;49(5):1594-1610. doi:10.15672/hujms.569410
Chicago Akgün, Ramazan. “Fractional Order Mixed Difference Operator and Its Applications in Angular Approximation”. Hacettepe Journal of Mathematics and Statistics 49, no. 5 (October 2020): 1594-1610. https://doi.org/10.15672/hujms.569410.
EndNote Akgün R (October 1, 2020) Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics 49 5 1594–1610.
IEEE R. Akgün, “Fractional order mixed difference operator and its applications in angular approximation”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, pp. 1594–1610, 2020, doi: 10.15672/hujms.569410.
ISNAD Akgün, Ramazan. “Fractional Order Mixed Difference Operator and Its Applications in Angular Approximation”. Hacettepe Journal of Mathematics and Statistics 49/5 (October 2020), 1594-1610. https://doi.org/10.15672/hujms.569410.
JAMA Akgün R. Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics. 2020;49:1594–1610.
MLA Akgün, Ramazan. “Fractional Order Mixed Difference Operator and Its Applications in Angular Approximation”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, 2020, pp. 1594-10, doi:10.15672/hujms.569410.
Vancouver Akgün R. Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics. 2020;49(5):1594-610.