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- Title
TAUBERIAN THEOREMS FOR THE MEAN OF LEBESGUE-STIELTJES INTEGRALS.
- Authors
Sendov, Hristo S.; Junquan Xiao
- Abstract
Suppose s(x): [a,∞) 7→ R is locally integrable with respect to a Radon measure μ on [a,∞). The mean of s(x) with respect to μ is defined to be ... where F(x) = μ(a, x]. A scallar l is called the statistical limit of s(x) as x → ∞ if for every ε > 0, ... This is denoted by st-lim x!1 s(x) = l. The following Tauberian theorems are proved under mild assymptotic conditions on F(t) and assuming that s(x) is slowly decreasing with respect to F(t). 1. If lim t!1 τ(t) = l, then lim x!1 s(x) = l. 2. If st-lim x!1 s(x) = l, then lim x!1 s(x) = l. 3. If st-lim t!1 τ(t) = l, then lim x!1 s(x) = l. This work extends results obtained by F. M'oricz and Z. N'emeth in [3] and [4] for the case F(t) = log(t).
- Subjects
TAUBERIAN theorems; LEBESGUE integral; RADON measures; STATISTICS; MATHEMATICAL sequences
- Publication
Serdica Mathematical Journal, 2017, Vol 43, Issue 3/4, p293
- ISSN
1310-6600
- Publication type
Article