Abstract
In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative estimates in terms of higher order moduli of smoothness and of the best approximation quantity. As a byproduct, we obtain two new results for complex Bloch and Besov spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Castillo Villalba, C.M.P., Colombo, F., Gantner, J., Gonzalez Cervantes, J.O.: Bloch, Besov and Dirichlet spaces of slice hyperholomorphic functions. Complex Anal. Oper. Theory 9, 479–517 (2015)
Colombo, F., González-Cervantes, J.O., Luna-Elizarrarás, M.E., Sabadini, I., Shapiro, M.: On two approaches to the Bergman theory for slice regular functions. Springer INdAM Ser. 1, 39–54 (2013)
Colombo, F., González-Cervantes, Sabadini I: Further properties of the Bergman spaces of slice regular functions. Adv. Geom. 15, 469–484 (2015)
Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhäuser/Springer Basel AG, Basel (2011)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Diki, K., Gal, S.G., Sabadini, I.: Polynomial approximation in slice regular Fock spaces. Complex Anal. Oper. Theory 13(6), 2729–2746 (2019)
Gal, S.G.: Quantitative approximations by convolution polynomials in Bergman spaces. Complex Anal. Oper. Theory 12(2), 355–364 (2018)
Gal, S.G.: Convolution-type integral operators in complex approximation. Comput. Methods Funct. Theory 1(2), 417–432 (2001)
Gal, S.G., Sabadini, I.: Approximation in compact balls by convolution operators of quaternion and paravector variable. Bull. Belg. Math. Soc. Simon Stevin 20, 481–501 (2013)
Gal, S.G., Sabadini, I.: Approximation by polynomials on quaternionic compact sets. Math. Methods Appl. Sci. 38(14), 3063–3074 (2015)
Gal, S.G., Sabadini, I.: Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball. Math. Methods Appl. Sci. 41(4), 1619–1630 (2018)
Gal, S.G., Sabadini, I.: Quaternionic Approximation. With Application to Slice Regular Functions. Frontiers in Mathematics. Springer, Berlin (2019)
Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer, Berlin (2013)
Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and \(n\)-Dimensional Space. Birkhäuser, Basel (2008)
Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000)
Lorentz, G.G.: Approximation of Functions. Chelsea Publ. Comp, New York (1986)
Triebel, H.: Theory of Function Spaces. Birkhäuser/Springer Basel AG, Basel (2010)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)
Zhu, K.: Analytic Besov spaces. J. Math. Anal. Appl. 157(3), 318–336 (1991)
Acknowledgements
The authors thank the reviewers for their useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the Topical Collection on ISAAC 12 at Aveiro, July 29–August 2, 2019, edited by Swanhild Bernstein, Uwe Kaehler, Irene Sabadini, and Franciscus Sommen.
Rights and permissions
About this article
Cite this article
Gal, S.G., Sabadini, I. Polynomial Approximation in Quaternionic Bloch and Besov Spaces. Adv. Appl. Clifford Algebras 30, 64 (2020). https://doi.org/10.1007/s00006-020-01084-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-020-01084-6