TY - JOUR

T1 - On the decay rate of (p, A)-lacunary series

AU - Nazarov, F. L.

AU - Shirokov, N. A.

N1 - Funding Information: Aknowledgments. This research was supported by the Russian Foundation for Basic Research (proect 05-01-0924).

PY - 2006/11

Y1 - 2006/11

N2 - A power series ∑k=0∞akx nk with radius of convergence equal 1 is called a (p,A)-lacunary one if nk ≥ Akp, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition |f(x)|exp ( B(1 - x)- 1/p - 1 + ε(1 - x) -1/p - 1/(|log (1 - x)| + 1)x→1-0→ 0, where B = (p - 1)(π/p)p/p - 1.1/A1/(p-1).1/|cos πp/2|1/(p-1), for some ε > 0, then f ≡0. We construct a (p,A)-lacunary series f 0 such that |f0 (x)|exp ( B(1 - x)-1/p-1 + C0(1 - x)-1/p-1/(|log(1 - x)|2 + 1)) x→1-0 → 0 with a constant C0 = C 0(p,A) > 0. Bibliography: 4 titles.

AB - A power series ∑k=0∞akx nk with radius of convergence equal 1 is called a (p,A)-lacunary one if nk ≥ Akp, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition |f(x)|exp ( B(1 - x)- 1/p - 1 + ε(1 - x) -1/p - 1/(|log (1 - x)| + 1)x→1-0→ 0, where B = (p - 1)(π/p)p/p - 1.1/A1/(p-1).1/|cos πp/2|1/(p-1), for some ε > 0, then f ≡0. We construct a (p,A)-lacunary series f 0 such that |f0 (x)|exp ( B(1 - x)-1/p-1 + C0(1 - x)-1/p-1/(|log(1 - x)|2 + 1)) x→1-0 → 0 with a constant C0 = C 0(p,A) > 0. Bibliography: 4 titles.

UR - http://www.scopus.com/inward/record.url?scp=33750162142&partnerID=8YFLogxK

U2 - 10.1007/s10958-006-0361-x

DO - 10.1007/s10958-006-0361-x

M3 - Article

AN - SCOPUS:33750162142

VL - 139

SP - 6437

EP - 6446

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -