TY - JOUR

T1 - Approximation by entire functions on subsets of a ray

AU - Sil'vanovich, O. V.

AU - Shirokov, N. A.

PY - 2007/6/1

Y1 - 2007/6/1

N2 - Let E ∈ ℝ+ be a set consisting of finitely many intervals and a ray [a,∞), and let H ω r be the set of functions defined on E for which |fr(x) - f(r) (y)| ≤cfω(|x - y|), where the continuity module ω(x) satisfies the condition ∫y oω(x)/x dx + y ∫∞yω(x)/x2dx ≤ C0ω(y), y > 0. Let C σ (r,ω) , r > 0, denote the class of entire functions F of order 1/2 and of type σ such that sup|F(z)|̇e-σ|Im √z|z∈C\ℝ (1 + |z|r ω (|z|) + σ -2r ω(σ-2) < <. In the paper, given a function f ∈ H ω r (E), we construct approximating functions F in the class C σ (r,ω) . Approximation by such functions on the set E is analogous to approximation by polynomials on compacts. The analogy involves constructing a scale for measuring approximations and providing a constructive description of the class H ω r (E) in terms of the approximation rate, similar to that of polynomial approximation. Bibliography: 4 titles.

AB - Let E ∈ ℝ+ be a set consisting of finitely many intervals and a ray [a,∞), and let H ω r be the set of functions defined on E for which |fr(x) - f(r) (y)| ≤cfω(|x - y|), where the continuity module ω(x) satisfies the condition ∫y oω(x)/x dx + y ∫∞yω(x)/x2dx ≤ C0ω(y), y > 0. Let C σ (r,ω) , r > 0, denote the class of entire functions F of order 1/2 and of type σ such that sup|F(z)|̇e-σ|Im √z|z∈C\ℝ (1 + |z|r ω (|z|) + σ -2r ω(σ-2) < <. In the paper, given a function f ∈ H ω r (E), we construct approximating functions F in the class C σ (r,ω) . Approximation by such functions on the set E is analogous to approximation by polynomials on compacts. The analogy involves constructing a scale for measuring approximations and providing a constructive description of the class H ω r (E) in terms of the approximation rate, similar to that of polynomial approximation. Bibliography: 4 titles.

UR - http://www.scopus.com/inward/record.url?scp=34248157742&partnerID=8YFLogxK

U2 - 10.1007/s10958-007-0198-y

DO - 10.1007/s10958-007-0198-y

M3 - Article

AN - SCOPUS:34248157742

VL - 143

SP - 3149

EP - 3152

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 3

ER -