Tbilisi Mathematical Journal

A characterization of weighted Besov spaces in quantum calculus

Akram Nemri and Belgacem Selmi

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Abstract

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.

Article information

Source
Tbilisi Math. J., Volume 9, Issue 1 (2016), 29-48.

Dates
Received: 16 September 2015
Accepted: 15 December 2015
First available in Project Euclid: 12 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1528769040

Digital Object Identifier
doi:10.1515/tmj-2016-0004

Mathematical Reviews number (MathSciNet)
MR3459009

Zentralblatt MATH identifier
1332.33029

Subjects
Primary: 33D60: Basic hypergeometric integrals and functions defined by them
Secondary: 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

Keywords
$q$-theory $q$-weighted Besov spaces $q$-Caldeón's formula $q$-convolution product

Citation

Nemri, Akram; Selmi, Belgacem. A characterization of weighted Besov spaces in quantum calculus. Tbilisi Math. J. 9 (2016), no. 1, 29--48. doi:10.1515/tmj-2016-0004. https://projecteuclid.org/euclid.tbilisi/1528769040


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References

  • L. D. Abreu, Functions $q$-orthogonal with respect to their own zeros, Proc. Amer. Math. Soc. 134(2006), 2695–2701.
  • G. E. Andrews, $q$-Series: their development in analysis number theory, combinatorics, physics and computer algebra, CBMS Series, Amer. Math. Soc. Providence, RI, \bf66(1986), 223–241.
  • J. L. Ansorna and O. Blasco, Characterization of weighted Besov spaces, Math. Nachr., \bf171(1995), 5–17
  • O.V. Besov, On a family of function spaces in connection with embeddings and extentions, Trudy Math. Inst. Steklov, 60 (1966), 42–81.
  • M. Bohner, M. Fan and J. Zhang, Periodicity of scalar dynamic equaton and application to population models, J. Math. Anal. appl. \bf330 (2007), 1–9.
  • M. Bohner, T. Hudson, Euler-type boundary value problems in quantum calculus , International Journal of Applied Mathematics and Statistics, \bf9(2007), 19–23.
  • 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.
  • G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of mathematics and its applications 35, Cambridge university press, 1990.
  • 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.
  • A. Fitouhi and F. Bouzeffour, $q$-cosine Fourier transform and $q$-heat equation, Ramanjuan J. in press.
  • A. Fitouhi, L. Dhaouadi and J. El Kamel, Inequalities in $q$-Fourier analysis , J. Inequal. Pure Appl. Math. 171(2006), 1–14.
  • A. Fitouhi, M. Hamza and F. Bouzeffour, The $q$-$J_{\alpha}$ Bessel function, J. Approx. Theory 115(2002), 114–116.
  • A. Fitouhi and A. Nemri, Distribution and convolution product in quantum calculus, Afr. Diaspora. J. Math, 7(2008), 39–58.
  • T.M. Flett, Lipschitz spaces of functions on the circle and the disc, J. Math. Anal. and appl, \bf39(1972), 125–158.
  • T. M. Flett, Temperatures, Bessel potentials and Lipschitz spaces, Proc. London Math. Soc, \bf20(1970), 749–768.
  • F.H. Jackson, On $q$-definite integrals, Quart. J. Pure. Appl. Math, 41(1910), 193–203.
  • V.G. Kac and P. Cheeung, Quantum calculus, Universitext, Springer-Verlag, New York, (2002).
  • A. Nemri and B. Selmi, Sobolev type spaces in quantum calculus, J. Math. Anal. Appl, 359(2009), 588–601.
  • A. Nemri and B. Selmi, On a Calderón's formula in quantum calculus, Indagationes Mathematicae, 24(2013), 491–504.
  • J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Series, NC,(1976).
  • S. Sanyal, Stochastic dynamic equation, PhD Dissertation, Missouri University of Science and Technology (2008).
  • 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, \bf7(2006), 395–413.
  • M. Taibleson, On the theory of Lipschitz spaces of distributions on euclidean n-space, I, II,III.
  • A. Torchinsky, Real-variable Methods in Harmonics Analysis, Academic Press, (1986).
  • H. Triebel, Theory of functon spaces, Monographs in Math., vol. 78, Birkäuser, Verlag, Basel, (1983).