]> 4.4 Discussion of results

4.4 Discussion of results

The existence and uniqueness problems of the best 2–approximation have been analysed. An important question is the error estimation as a function of n. In the approximation theory such estimates are known as the Jackson–type inequalities. Since the maximum of J at Sn is greater than the maximum of J at "the diagonal" λ1 = λ2 = = λn, then one of possible ways to get a Jackson–type inequality is to estimate the approximation error with the aid of the Laguerre functions, which follows from Remark 4.2.1. Such an idea was suggested also in [27, Section 6]. A derivation of a Jackson–type inequality requires considering the countable sequence {λi}i=1 and thus it is naturally close to the question whether the system of exponentials {eλit} i=1 forms Riesz basis in L2(0,) [1], [23], [67].

Starting from the semigroups of shifts (or even weighted shifts) generators in other spaces, related to L2(0,), one can get generalizations of the above results. For instance, in the space 2(), = {0, 1, 2,} the semigroup of left–shifts generator has the form

Ax(j) = x(j + 1),j = 0, 1, 2, (4.17)

This is a linear and bounded operator with the point spectrum σP (A) = D , D denotes the interior of a unit circle. The k–th normalized eigenfunction of A, corresponding to λk σP (A) is fk(λk)(j) = 1 λk 2λkj, j = 0, 1, 2,. Here, we assume that λk0. If λk = 0 then the vectors of the Cartesian basis in 2() are generalized eigenvectors of A. Instead of the Hardy space H2(Π+) one should consider the Hardy space H2(𝔻) [25], [49], [67]. The formula (4.6) is still valid (see also [67, IX-4]) with the Malmquist functions being now of the form

ê1(λ,z) = 1 λ1 2 1 λ1z ,êk(λ,z) = 1 λk 2 1 λkz i=1k1 z λi¯ 1 λiz k = 2, 3,,n,λ = (λ1,λ2,,λn) (4.18)

(these are the Taylor transforms of a system {ek(λ)}k=1n resulting from the system {fk(λk)}k=1n, by application of the Gram–Schmidt orthonormalization). The representation of the k–th Fourier coefficient looks as follows

f,ek(λ) = 1 2πππf̂(ejθ)ê k(λ,ejθ)¯dθ = 1 2πj z=1f̂(z)êk λ, 1 z¯1 zdz = = i=1k Res z=λi¯ f̂(z)1 λk 2 z λk¯ i=1k11 λiz z λi¯ (4.19)

where f̂, êk denote the Taylor transforms of f, ek, respectively. Erokhin [24], using the other representation for the Fourier coefficients, reduced formally the best 2() – approximation problem with polynomials (equivalently the best H2(𝔻) – approximation problem with rational functions) to the maximization of the function

J : 𝔻n λ k=1n f,e k(λ)2 .

The proof technique worked out in Section 4.2, after some obvious modification allow us to establish that the maximum of J is attained not only on 𝔻n but also on Sn, where S is a relatively closed subset of 𝔻 (as in Theorem 4.1.1).

The existence of a maximum of J on 𝔻n has been proved also in [74], [2] with the aid of yet another proof technique also taking advantage of the Hardy spaces, however without any explicit expressions for J.