Ukrainian Mathematical Journal, Vol. 60, No. 5, 2008 BEST LINEAR METHODS OF APPROXIMATION AND OPTIMAL ORTHONORMAL SYSTEMS OF THE HARDY SPACE V. V. Savchuk UDC 517.5 We construct the best linear methods of approximation for functions of the Hardy space H p on compact subsets of the unit disk. It is shown that the Takenaka–Malmquist systems are optimal systems of functions orthonormal on the unit circle for the construction of the best linear meth- ods of approximation. 1. Notation. Statement of the Problem Let 1 ≤ p ≤ ∞ and let H p be the Hardy space of functions f holomorphic in the unit disk D : = z z ∈ < { } C : 1 with the finite norm f H p = sup ( ) ( ) , , sup () / 0 1 1 1 < < ∈ ∫ ( ) ≤ <∞ ρ ρ σ f w d w p fz p p z T D , , p =∞ ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ where T = z z ∈ = { } C : 1 is the unit circle, σ is the normalized Lebesgue measure on the circle T , and UH p is the unit ball in the space H p , i.e., UH p = f H f p ∈ ≤ { } : 1. It is known that every function f from the space H p has angular boundary values (denoted by the same letter f ) almost everywhere on the circle T, and, furthermore, f L p ∈ () T : = f f f d L p p p is summable on T T T : : ( ) / = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ <∞ ⎧ ∫ σ 1 ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ . Let a : = a k k { } = ∞ 0 be a sequence of points in the disk D among which there may be points of finite or even infinite multiplicity. A system ϕ : = ϕ k k { } = ∞ 0 of functions ϕ k of the form ϕ 0 () z = 1 1 0 2 0 – a az − , ϕ k z () = 1 1 1 2 0 1 – – – – – – a az a a z a az k k j j j k j j = ∏ , k = 1, 2, … , (1) Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 636–646, May, 2008. Original article submitted August 13, 2007. 730 0041–5995/08/6005–0730 © 2008 Springer Science+Business Media, Inc.