Approximation of Derivatives of Analytic Functions from One Hardy Class by Another Hardy Class

R. R. Akopyan

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 21-29

Résumé

In the Hardy space $\mathcal{H}^p(D_\varrho)$, $1\le p\le\infty$, of functions analytic in the disk $D_\varrho=\left\{z\in\mathbb{C}\,:\,|z|\varrho\right\}$, we denote by $NH^p(D_\varrho)$, $N>0$, the class of functions whose $L^p$-norm on the circle $\gamma_\varrho=\left\{z\in\mathbb{C} :\, |z|=\varrho\right\}$ does not exceed the number $N$ and by $\partial H^p(D_\varrho)$ the class consisting of the derivatives of functions from $1H^p(D_\varrho)$. We consider the problem of the best approximation of the class $\partial H^p(D_\rho)$ by the class $NH^p(D_R)$, $N>0$, with respect to the $L^p$‐norm on the circle $\gamma_r$, $0$. The order of the best approximation as $N\rightarrow+\infty$ is found: $$ \mathcal{E}\left(\partial H^p(D_\rho), NH^p(D_R)\right)_{L^p(\Gamma_r)} \asymp N^{-\beta/\alpha} \ln^{1/\alpha}N, \quad \alpha=\frac{\ln R-\ln\rho}{\ln R-\ln r}, \quad \beta=1-\alpha.$$ In the case where the parameter $N$ belongs to some sequence of intervals, the exact value of the best approximation and a linear method implementing it are obtained. A similar problem is considered for classes of functions analytic in annuli.

Détail

Keywords: analytic functions, Hardy class, best approximation of a class by a class.

Bibliographie

[1] Akopyan R.R., “Nailuchshee priblizhenie operatora analiticheskogo prodolzheniya na klasse analiticheskikh v koltse funktsii”, Tr. Instituta matematiki i mekhaniki UrO RAN, 18:4 (2012), 3–13

[2] Akopyan Roman R., “Approximation of the differentiation operator on the class of functions analytic in an annulus”, Ural Math. J., 3:2 (2017), 6–13 | DOI | MR

[3] Arestov V.V., “Priblizhenie lineinykh operatorov i rodstvennye ekstremalnye zadachi”, Tr. MIAN SSSR, 138, 1975, 29–42 | Zbl

[4] Arestov V.V., “O nekotorykh ekstremalnykh zadachakh dlya differentsiruemykh funktsii odnoi peremennoi”, Tr. MIAN SSSR, 138, 1975, 3–28 | Zbl

[5] Arestov V.V., “Nailuchshee priblizhenie odnogo klassa funktsii mnogikh peremennykh drugim i rodstvennye ekstremalnye zadachi”, Matem. zametki, 64:3 (1998), 323–340 | DOI | MR | Zbl

[6] Arestov V.V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, Uspekhi mat. nauk, 51:6 (312) (1996), 89–124 | DOI | MR | Zbl

[7] Taikov L.V., “Analiticheskoe prodolzhenie funktsii s oshibkoi”, Tr. MIAN SSSR, 109, 1971, 61–64 | Zbl

[8] Shvedenko S.V., “Klassy Khardi i svyazannye s nimi prostranstva analiticheskikh funktsii v edinichnom kruge, polikruge i share”, Itogi nauki i tekhniki. Ser. Mat. analiz, 23 (1985), 3–124 | MR | Zbl