Extremal Problem On Space Of Analytic Function

In this paper, we mainly discuss extremal problem of Hardy space Hl(Tn) of n-torus and the extremal function in weighted Bergman space which satisfied the given interpolation condition.In chapter one, we give a simple introduction to this paper, which include the background of research and main results of this paper.In chapter two, we discuss the extremal problem:forΦ∈L∞(Tn), define the linear functional BΦonH1(Tn) BΦ(f)＝∫Tnf(ζ)Φ(ζ)dm(ζ), whether there is a f∈H1(Tn), such that |BΦ(f)|=‖BΦ‖. We discuss the existence of the solution of above extremal problem, show that the collection of the solution of the extremal problem is non-void and weak-* compact whenΦis continue function on n torus, and we character the solution of the extremal problem whenΦis rational function or rational inner function or multiply of one variable functions.In chapter three, we discuss the extremal problem below:Give n independent continuous linear functional l1,l2,...ln on weighted Bergman space Aαp and n nonzero complex numbers c1,c2,...cn., Whether there is a non-vanishing f which belong to the Bergman space Aαp attain the below infimumλα= inf{‖f‖Aαp:f(z)≠0,li(f)=ci,1≤i≤n}. we show the existence of the solution and stability of extremal values when any functions which satisfied the interpolation condition exist.