Characterization of mixed modulus of smoothness in weighted \(L^p\) spaces

Ramazan Akgün

doi:10.33993/jnaat452-1094

Authors

Ramazan Akgün Balikesir University, Turkey

DOI:

https://doi.org/10.33993/jnaat452-1094

Keywords:

modulus of smoothness, Lebesgue spaces, Muckenhoupt weights, Fourier series, partial de la Vallee Poussin means, Lp spaces

Abstract views: 397

Abstract

Characterization class for mixed modulus of smoothness in Lebesgue spaces with Muckenhoupt weights are investigated.

Downloads

Download data is not yet available.

Author Biography

Ramazan Akgün, Balikesir University, Turkey

Faculty ofn Arts and Sciences, Department of Mathematics

References

R. Akgun, Realization and characterization of modulus of smoothness in weighted Lebesgue spaces, Algebra i Analiz, 26, pp. 64-87, 2014; St. Petersburg Math. J., 26 (2015), pp. 741-756, http://dx.doi.org/10.1090/spmj/1356

R. Akgun, Mixed modulus of continuity in Lebesgue spaces with Muckenhoupt weights, In press, Turkish Math. J., 40, 2016, http://dx.doi.org/10.3906/mat-1507-92 DOI: https://doi.org/10.3906/mat-1507-92

E. Berkson and T.A. Gillespie, On restrictions of multipliers in weighted setting, Indiana Univ. Math. J., 52 (2003), pp. 927-961, http://dx.doi.org/10.1512/iumj.2003.52.2368 DOI: https://doi.org/10.1512/iumj.2003.52.2368

O.V. Besov and S.B. Stechkin, A description of the moduli of continuity in L2, Proc. Steklov Inst. Math., 134 (1977), pp. 27-30.

P.L. Butzer, H. Dyckhoff, E. Gorlich and R. L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes, Canad. J. Math., 29 (1977), pp. 781-793, http://dx.doi.org/10.4153/cjm-1977-081-6 DOI: https://doi.org/10.4153/CJM-1977-081-6

C. Cottin, Mixed K-functionals: a measure of smoothness for blending-type approximation, Math. Z., 204 (1990), pp. 69-83 http://dx.doi.org/10.1007/bf02570860 DOI: https://doi.org/10.1007/BF02570860

R. Fefferman and E. Stein, Singular integrals in product spaces, Adv. Math., 45 (1982), pp. 117–143, http://dx.doi.org/10.1016/s0001-8708(82)80001-7 DOI: https://doi.org/10.1016/S0001-8708(82)80001-7

A. Guven and V. Kokilashvili, On the mean summability by Cesaro method of Fourier trigonometric series in two-weighted setting, J. Inequal. Appl., bf 2006, Article ID: 41837 (2006), pp. 1-15, http://dx.doi.org/10.1155/jia/2006/41837 DOI: https://doi.org/10.1155/JIA/2006/41837

R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc.,176 (1973), pp. 227-251, http://dx.doi.org/10.2307/1996205 DOI: https://doi.org/10.1090/S0002-9947-1973-0312139-8

D.M. Israfilov, Approximation by p-Faber Polynomials in the Weighted Smirnov Class Ep(G,ω) and the Bieberbach Polynomials, Constr. Approx.,17 (2001), pp. 335-351, http://dx.doi.org/10.1007/s003650010030 DOI: https://doi.org/10.1007/s003650010030

S.Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 33 (2010), pp. 95-100.

V.I. Kolyada, Imbedding in the classes φ(L), Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), pp. 418-437.

A.D. Nakhman and B.P. Osilenker, Estimates of weighted norms of some operators generated by multiple trigonometric Fourier series, Izvestiya Vysshikh Uchebnykh Zavedeni Matematika239 (1982), pp. 39–50 (Russian).

M. K. Potapov, Approximation by ”angle” (in Russian), In: Proceedings of the Conference on the Constructive Theory of Functions and Approximation Theory, Budapest, 1969; Akad ́emiai Kiad ́o, pp. 371-399, 1972.

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 (1972), pp. 339-362.

M. K. Potapov, A certain imbedding theorem in Russian, Mathematica (Cluj), 14 (1972), pp. 123-146.

M. K. Potapov, Approximation ”by angle”, and imbedding theorems, in Russian, Math. Balkanica, 2 (1972), pp. 183-198.

M. K. Potapov,Imbedding of classes of functions with a dominating mixed modulus of Smoothness in Russian, Trudy Mat. Inst. Steklov., 131 (1974), pp. 199-210.

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 (1996), pp. 243-259.

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 (1995), pp. 167-192.

M.K. Potapov, B. V. Simonov and S. Y. Tikhonov, Embedding theorems for Besov-Nikolskiĭ and Weyl-Nikolskiĭ classes in a mixed metric, Moscow Univ. Math. Bull., 59 (2005), pp. 19-26.

M.K. Potapov, B. V. Simonov and S. Y. Tikhonov, Transformation of Fourier series using power and weakly oscillating sequences, Math. Notes., 77 (2005), pp. 90-107, http://dx.doi.org/10.1007/s11006-005-0009-z DOI: https://doi.org/10.1007/s11006-005-0009-z

M.K. Potapov, B. V. Simonov and S. Y. Tikhonov, Relations between mixed moduli of smoothness and embedding theorems for the Nikolskiĭ classes , Proc. Steklov Inst. Math., 269 (2010), pp. 197-207, http://dx.doi.org/10.1134/s0081543810020173 DOI: https://doi.org/10.1134/S0081543810020173

M.K. Potapov, B. V. Simonov and S. Y. Tikhonov, Mixed moduli of smoothness in Lp,1< p <∞: A survey, Surv. Approx. Theory, 8 (2013), pp. 1-57.

K.V. Runovski, Several questions of approximation theory, PhD Disser. Cand. Nauk., Moscow State University MGU, Moscow, Russia, 1989.

S. Tikhonov, On moduli of smoothness of fractional order, Real Anal. Exchange., 30, pp. 1-12, (2004/2005). DOI: https://doi.org/10.14321/realanalexch.30.2.0507

A.F. Timan, Theory of approximation of functions of a real variable, London: Pergamon Press, 1963. DOI: https://doi.org/10.1016/B978-0-08-009929-3.50008-7

M. F. Timan, Approximation and properties of periodic functions, Dnepropetrovsk: ”Fedorchenko”, 2011.

Y. E. Yildirir and D.M. Israfilov, Approximation theorems in weighted Lorentz spaces, Carpathian J. Math., 26 (2010), pp. 108-119