Bounded Extremal Problems in Bergman and Bergman-Vekua spaces

Briceyda B Delgado (1) , Juliette Leblond (2)

1 SFEDU - Southern Federal University [Rostov-on-Don]
2 FACTAS - Analyse fonctionnelle pour la conception et l'analyse de systèmes

We analyze Bergman spaces A p f (D) of generalized analytic functions of solutions to the Vekua equation ∂w = (∂f /f)w in the unit disc of the complex plane, for Lipschitz-smooth non-vanishing real valued functions f and 1 < p < ∞. We consider a family of bounded extremal problems (best constrained approximation) in the Bergman space A p (D) and in its generalized version A p f (D), that consists in approximating a function in subsets of D by the restriction of a function belonging to A p (D) or A p f (D) subject to a norm constraint. Preliminary constructive results are provided for p = 2.

Mots clés: Generalized holomorphic functions; pseudo-holomorphic / pseudo-analytic functions; Bergman spaces; Vekua equation; Bergman-Vekua spaces; bounded extremal problems Classification: 30G20; 30H20; 30E10; 41A29

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]; Variables complexes [math.CV]

Briceyda B Delgado, Juliette Leblond. Bounded Extremal Problems in Bergman and Bergman-Vekua spaces. Complex Variables and Elliptic Equations, inPress, ⟨10.1080/17476933.2020.1818732⟩. ⟨hal-02928903⟩