Abstract
We show that there exists a de Branges–Rovnyak space \({\mathcal {H}}(b)\) on the unit disk containing a function f with the following property: even though f can be approximated by polynomials in \({\mathcal {H}}(b)\), neither the Taylor partial sums of f nor their Cesàro, Abel, Borel or logarithmic means converge to f in \({\mathcal {H}}(b)\). A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Boos, J.: Classical and modern methods in summability. Oxford Mathematical Monographs. Oxford University Press, Oxford. Assisted by Peter Cass, Oxford Science Publications (2000)
Borwein, D.: On a scale of Abel-type summability methods. Proc. Camb. Philos. Soc. 53, 318–322 (1957)
Borwein, D.: On methods of summability based on power series. Proc. Roy. Soc. Edinb. Sect. A 64, 342–349 (1957)
Borwein, D., Watson, B.: On the relation between the logarithmic and Borel-type summability methods. Canad. Math. Bull. 24(2), 153–159 (1981)
Chevrot, N., Guillot, D., Ransford, T.: De Branges–Rovnyak spaces and Dirichlet spaces. J. Funct. Anal. 259(9), 2366–2383 (2010)
Duren, P.L.: Theory of \(H^p\) Spaces. Pure and Applied Mathematics. Academic Press, New York and London (1970)
El-Fallah, O., Fricain, E., Kellay, K., Mashreghi, J., Ransford, T.: Constructive approximation in de Branges–Rovnyak spaces. Constr. Approx. 44(2), 269–281 (2016)
Hardy, G.H.: Divergent series. Clarendon Press, Oxford (1949)
Hille, E., Phillips, R. S.: Functional analysis and semi-groups. In: American Mathematical Society, Providence, R. I. Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications, Vol. XXXI (1974)
Jevtić, M., Vukotić, D., Arsenović, M.: Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces. RSME Springer Series, vol. 2. Springer, Cham (2016)
Mashreghi, J., Parisé, P.-O., Ransford, T.: Failure of approximation of odd functions by odd polynomials. Constr. Approx. to appear
Mashreghi, J., Ransford, T.: Linear polynomial approximation schemes in banach holomorphic function spaces. Anal. Math. Phys. 9, 899–905 (2019)
Robinson, A.: On functional transformations and summability. Proc. Lond. Math. Soc. 2(52), 132–160 (1950)
Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. The University of Arkensas Lecture Notes in the Mathematical Sciences. Wiley, Hoboken (1994)
Acknowledgements
We are grateful to the anonymous referee for reading the article carefully and for making several helpful suggestions to improve the exposition.
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.
JM supported by an NSERC Discovery Grant. POP supported by an NSERC Alexander-Graham-Bell Scholarship and a scholarship from FRQNT. TR supported by grants from NSERC and the Canada Research Chairs program.
Rights and permissions
About this article
Cite this article
Mashreghi, J., Parisé, PO. & Ransford, T. Power-Series Summability Methods in de Branges–Rovnyak Spaces. Integr. Equ. Oper. Theory 94, 20 (2022). https://doi.org/10.1007/s00020-022-02698-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00020-022-02698-0
Keywords
- De Branges–Rovnyak spaces
- Polynomial approximations
- Summability methods
- Logarithmic means
- Abel means
- Borel means