Also we obtain an equivalence between fractional mixed modulus of smoothness and fractional mixed K-functional.********************************************************************************************************************
📌 3 Maddede Bu Makale
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Amaç & Hipotez:Lebesgue spaces are considered with Muckenhoupt weights.
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Temel Bulgular & Veri:Fractional order mixed difference operator is investigated to obtain mixed fractional modulus of smoothness in these spaces.
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Sonuç & Katkı:Also we obtain an equivalence between fractional mixed modulus of smoothness and fractional mixed K-functional.********************************************************************************************************************
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.********************************************************************************************************************
Akgün, R. (2022). Fractional order mixed difference operator and its applications in angular approximation. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 5, 1594–1610. https://ajindex.com/articles/529672
R. Akgün, Polynomial approximation in weighted Lebesgue spaces, East J. Approx. 17(3), 253–266, 2011
R. Akgün, Approximating polynomials for functions of weighted Smirnov-Orlicz spaces, J. Funct. Spaces Appl. 2012, 1–41, ID 982360, 2012
R. Akgün, Realization and characterization of modulus of smoothness in weighted Lebesgue spaces, St. Petersburg Math. J. 26 (5), 741–756, 2015
R. Akgün, Mixed modulus of continuity in Lebesgue spaces with Muckenhoupt weights, Turkish J. Math. 40 (6), 1169–1192, 2016
R. Akgün, Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle, Complex Var. Elliptic Equ. 64 (2), 330–351, 2019
R. Akgün, Nikol’ski, Jackson and Ul’yanov type inequalities with Muckenhoupt weights, arXiv:1709.02928 [math.CA]
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
C. Cottin, Mixed K-functionals: a measure of smoothness for blending-type approximation, Math. Z. 204 (1), 69–83, 1990
E. DiBenedetto, Real Analysis, Second Ed., Birkhäuser, Boston, 2016
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
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
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
S.Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 33, 95–100, 2010
V. Kokilashvili and Y.E. Yildirir, On the approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst. 143, 103–113, 2007
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
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
M.K. Potapov, The Hardy-Littlewood and Marcinkiewicz-Littlewood-Paley theorems, approximation "by an angle”, and the imbedding of certai