# zbMATH — the first resource for mathematics

A characterization of weighted Besov spaces in quantum calculus. (English) Zbl 1332.33029
Summary: In this paper, subspaces of $$L^{p}(\mathbb{R}_{q,+})$$ are defined using $$q$$-translations $$T_{q,x}$$ operator and $$q$$-differences operator, called $$q$$-Besov spaces. We provide characterization of these spaces by using the $$q$$-convolution product.

##### MSC:
 33D60 Basic hypergeometric integrals and functions defined by them 26D15 Inequalities for sums, series and integrals 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 33D90 Applications of basic hypergeometric functions
Full Text:
##### References:
 [1] L. D. Abreu, Functions q-orthogonal with respect to their own zeros, Proc. Amer. Math. Soc. 134(2006), 2695-2701. · Zbl 1091.33013 [2] G. E. Andrews, q-Series: their development in analysis number theory, combinatorics, physics and computer algebra, CBMS Series, Amer. Math. Soc. Providence, RI, 66(1986), 223-241. [3] J. L. Ansorna and O. Blasco, Characterization of weighted Besov spaces, Math. Nachr., 171(1995), 5-17 · Zbl 0813.46023 [4] O.V. Besov, On a family of function spaces in connection with embeddings and extentions, Trudy Math. Inst. Steklov, 60 (1966), 42-81. [5] M. Bohner, M. Fan and J. Zhang, Periodicity of scalar dynamic equaton and application to population models, J. Math. Anal. appl. 330 (2007), 1-9 . · Zbl 1179.34106 [6] M. Bohner, T. Hudson, Euler-type boundary value problems in quantum calculus , International Journal of Applied Mathematics and Statistics, 9(2007), 19-23. · Zbl 1144.39010 [7] L. Dhaouadi, J. El Kamel and A. Fitouhi, Positivity of q-even translation and Inequality in q-Fourier analysis, JIPAM. J. Inequal. Pure Appl. Math 171(2006), 1-14. · Zbl 1232.26025 [8] G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of mathematics and its applications 35, Cambridge university press, 1990. [9] I. Gravagne, J. Davis and R. Marks II, How deterministic must a real time controller be, Proceedings of 2005, IEEE/RSI International Conference on intelligent Robots and Systems, Alberta, Aug. 2-6, (2005), 3856-3861. [10] A. Fitouhi and F. Bouzeffour, q-cosine Fourier transform and q-heat equation, Ramanjuan J. in press. · Zbl 1255.33009 [11] A. Fitouhi, L. Dhaouadi and J. El Kamel, Inequalities in q-Fourier analysis , J. Inequal. Pure Appl. Math. 171(2006), 1-14. · Zbl 1232.26025 [12] A. Fitouhi, M. Hamza and F. Bouzeffour, The q-J_{α} Bessel function, J. Approx. Theory 115(2002), 114-116. · Zbl 1003.33007 [13] A. Fitouhi and A. Nemri, Distribution and convolution product in quantum calculus, Afr. Diaspora. J. Math, 7(2008), 39-58. · Zbl 1209.33013 [14] T.M. Flett, Lipschitz spaces of functions on the circle and the disc, J. Math. Anal. and appl, 39(1972), 125-158. · Zbl 0253.46084 [15] T. M. Flett, Temperatures, Bessel potentials and Lipschitz spaces, Proc. London Math. Soc, 20(1970), 749-768. · Zbl 0211.39203 [16] F.H. Jackson, On q-definite integrals, Quart. J. Pure. Appl. Math, 41(1910), 193-203. · JFM 41.0317.04 [17] V.G. Kac and P. Cheeung, Quantum calculus, Universitext, Springer-Verlag, New York, (2002). [18] A. Nemri and B. Selmi, Sobolev type spaces in quantum calculus, J. Math. Anal. Appl, 359(2009), 588-601. · Zbl 1187.46030 [19] A. Nemri and B. Selmi, On a Calderón’s formula in quantum calculus, Indagationes Mathematicae, 24(2013), 491-504. · Zbl 1290.33018 [20] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series, NC,(1976). · Zbl 0356.46038 [21] S. Sanyal, Stochastic dynamic equation, PhD Dissertation, , Missouri University of Science and Technology (2008). [22] Q. Sheng, M. Fadag, J. Henderson and J. Davis, An exploiration of combined dynamic derivatives on time scales and their applications, Nonlinear Analysis: Real World Applications, 7(2006), 395-413. · Zbl 1114.26004 [23] M. Taibleson, On the theory of Lipschitz spaces of distributions on euclidean n-space, I, II,III. · Zbl 0173.16103 [24] A. Torchinsky, Real-variable Methods in Harmonics Analysis, Academic Press , (1986). · Zbl 0621.42001 [25] H. Triebel, Theory of functon spaces, Monographs in Math., vol. 78, Birkuser, Verlag, Basel, (1983).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.