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 Mat. Zametki, 2002, Volume 72, Issue 5, Pages 750–764 (Mi mz465)

On the Structure of Spaces of Polyanalytic Functions

A.-R. K. Ramazanov

Kaluga Branch of Bauman Moscow State Technical University

Abstract: Suppose that $A_mL_p(D,\alpha)$ is the space of all $m$-analytic functions on the disk $D=ż:|z|<1\}$ which are $p$th power integrable over area with the weight $(1-|z|^2)^\alpha$, $\alpha >-1$. In the paper, we introduce subspaces $A_kL_p^0(D,\alpha)$, $k=1,2,…,m$, of the space $A_mL_p(D,\alpha)$ and prove that $A_mL_p(D,\alpha)$ is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.

DOI: https://doi.org/10.4213/mzm465

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English version:
Mathematical Notes, 2002, 72:5, 692–704

Bibliographic databases:

UDC: 517.5
Revised: 16.10.2001

Citation: A.-R. K. Ramazanov, “On the Structure of Spaces of Polyanalytic Functions”, Mat. Zametki, 72:5 (2002), 750–764; Math. Notes, 72:5 (2002), 692–704

Citation in format AMSBIB
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\jour Math. Notes
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\pages 692--704
\crossref{https://doi.org/10.1023/A:1021469308636}
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• http://mi.mathnet.ru/eng/mz465
• https://doi.org/10.4213/mzm465
• http://mi.mathnet.ru/eng/mz/v72/i5/p750

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This publication is cited in the following articles:
1. A. K. Ramazanov, “Estimate of the Norm of a Polyanalytic Function via the Norm of Its Polyharmonic Component”, Math. Notes, 75:4 (2004), 568–573
2. Du, JY, “Mixed boundary value problem for some pairs of metaanalytic function and analytic function”, Mathematical Methods in the Applied Sciences, 31:15 (2008), 1761
3. Wang Y., Du J.Yu., “Haseman Boundary Value Problems for Metaanalytic Functions with Different Shifts on the Unit Circumference”, Complex Var. Elliptic Equ., 53:4 (2008), 325–342
4. Wang Y., Du J., “On Haseman boundary value problem for a class of metaanalytic functions with different factors on the unit circumference”, Mathematical Methods in the Applied Sciences, 33:5 (2010), 576–584
5. Cuckovic Z., Le T., “Toeplitz Operators on Bergman Spaces of Polyanalytic Functions”, Bull. London Math. Soc., 44:Part 5 (2012), 961–973
6. Abreu L.D., Groechenig K., “Banach Gabor Frames with Hermite Functions: Polyanalytic Spaces From the Heisenberg Group”, Appl. Anal., 91:11 (2012), 1981–1997
7. V. I. Danchenko, “Cauchy and Poisson formulas for polyanalytic functions and applications”, Russian Math. (Iz. VUZ), 60:1 (2016), 11–21
8. Daghighi A., “A Necessary Condition For Weak Maximum Modulus Sets of 2-Analytic Functions”, Collect. Math., 69:2 (2018), 173–180
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