A presentation of extremal functions in the explicit form for the wide class of linear functionals on H1 - space

In this work extremal functions for the general class of functionals above Hardy space H1 was founded in the explicitly form. Linear functional l ( x) ∈ 2πH ∗ , which is given by the form l ( x) = 12π∫ x(e iθ )ω 0 (e iθ ) dθ, x ∈H 1 , x(0) = 0 , where ω ( z ) ∈ Lipα ∩ H ∞ . This article proves that the extremal function of the above-mentioned task can be presented in the following form: f (t ) = Rt ⋅ ∑RCk ϕ k (t )∑Ck ϕ k (t ) , t = 1 .k = 1k = 1 where R ≤ κ, κ - order of the largest positive root of the Fredholm`s deter- minant D(λ ) ,C 1 , C 1 , …, 1 and C 2 , C 2 , …, 2 - certain complex numbers, and ϕ k ∈ Lipα ∩ H ∞ - functions, defined in the Fed- holm`s minor form for integral equation( ) = λ ∫( ) ω (t ) − ω (ξ ) Υ ( )Y ξ ω t2π i t =1t − ξ t dt , ξ = 1 .