Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 1, page 305-317
  • ISSN: 0011-4642

Abstract

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Using partial derivatives ∂ f / ∂ z and ∂ f / ∂ z ¯ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree q can be approximated in norm by polyanalytic polynomials of degree at most q .

How to cite

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Abkar, Ali. "Polyanalytic Besov spaces and approximation by dilatations." Czechoslovak Mathematical Journal 74.1 (2024): 305-317. <http://eudml.org/doc/299221>.

@article{Abkar2024,
abstract = {Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar\{z\}$, we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$.},
author = {Abkar, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {mean approximation; polyanalytic Besov space; polyanalytic Bergman space; dilatation; non-radial weight; angular weight},
language = {eng},
number = {1},
pages = {305-317},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Polyanalytic Besov spaces and approximation by dilatations},
url = {http://eudml.org/doc/299221},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Abkar, Ali
TI - Polyanalytic Besov spaces and approximation by dilatations
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 1
SP - 305
EP - 317
AB - Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$, we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$.
LA - eng
KW - mean approximation; polyanalytic Besov space; polyanalytic Bergman space; dilatation; non-radial weight; angular weight
UR - http://eudml.org/doc/299221
ER -

References

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  3. Abkar, A., 10.1007/s13324-022-00671-z, Anal. Math. Phys. 12 (2022), Article ID 52, 16 pages. (2022) Zbl1486.30130MR4396660DOI10.1007/s13324-022-00671-z
  4. Abreu, L. D., Feichtinger, H. G., 10.1007/978-3-319-01806-5_1, Harmonic and Complex Analysis and its Applications Trends in Mathematics. Springer, Cham (2014), 1-38. (2014) Zbl1318.30070MR3203099DOI10.1007/978-3-319-01806-5_1
  5. Balk, M. B., Polyanalytic Functions, Mathematical Research 63. Akademie, Berlin (1991). (1991) Zbl0764.30038MR1184141
  6. Duren, P., Gallardo-Gutiérrez, E. A., Montes-Rodríguez, A., 10.1112/blms/bdm026, J. Lond. Math. Soc. 39 (2007), 459-466. (2007) Zbl1196.30046MR2331575DOI10.1112/blms/bdm026
  7. Haimi, A., Hedenmalm, H., 10.1016/j.jfa.2014.09.002, J. Funct. Anal. 267 (2014), 4667-4731. (2014) Zbl1310.30040MR3275106DOI10.1016/j.jfa.2014.09.002
  8. Košelev, A. D., On the kernel function of the Hilbert space of functions polyanalytic in a disk, Dokl. Akad. Nauk SSSR 232 (1977), 277-279 Russian. (1977) Zbl0372.30034MR0427648
  9. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Nauka, Moscow (1966), Russian. (1966) Zbl0151.36201MR0202367
  10. Ramazanov, A. K., 10.1023/A:1021469308636, Math. Notes 72 (2002), 692-704. (2002) Zbl1062.30055MR1963139DOI10.1023/A:1021469308636
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