TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

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dc.authoridOzsarac, Firat/0000-0001-7170-9613
dc.contributor.authorOzsarac, Firat
dc.contributor.authorCanak, Ibrahim
dc.date.accessioned2025-01-21T16:44:50Z
dc.date.available2025-01-21T16:44:50Z
dc.date.issued2020
dc.departmentKırıkkale Üniversitesi
dc.description.abstractLet q be a positive weight function on R+ := [0, infinity) which is integrable in Lebesgue's sense over every finite interval (0, x) for 0 < x < infinity, in symbol: q is an element of L-loc(1)(R+) such that Q(x) := integral(x)(0) q(t)dt not equal 0 for each x > 0, Q(0) = 0 and Q(x) -> infinity as x -> infinity. Given a real or complex-valued function f is an element of L-loc(1) (R+), we define s(x) := integral(x)(0) f (t)dt and tau((0))(q) (x) := s(x), tau((m))(q) (x) := 1/Q(x) integral(x)(0) tau((m-1))(q) (t)q(l)di (x > 0, m = 1, 2, ...), provided that Q(x) > 0, We say that integral(infinity)(0) (x)dx is summable to L by the m-th iteration of weighted mean method determined by the function q(x), or for short, ((N) over bar, q, m) integrable to a finite number L if x ->infinity(lim) tau((m))(q) (x) = L. In this case, we write s(x) -> L((N) over bar, q, m). It is known that if the limit lim(x ->infinity) (x) = L exists, then lim(x ->infinity) tau((m))(q) (x) = L also exists. However, the converse of this implication is not always true. Some suitable conditions together with the existence of the limit lim(x ->infinity) tau((m))(q) (x), which is so called Tauberian conditions, may imply convergence of lim(x ->infinity) s(x). In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for ((N) over bar, q , m) summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Cesaro summability (C, 1) and weighted mean method of summability ((N) over bar, q) have been extended and generalized.
dc.identifier.doi10.22190/FUM12003775O
dc.identifier.endpage788
dc.identifier.issn0352-9665
dc.identifier.issn2406-047X
dc.identifier.issue3
dc.identifier.startpage775
dc.identifier.urihttps://doi.org/10.22190/FUM12003775O
dc.identifier.urihttps://hdl.handle.net/20.500.12587/25527
dc.identifier.volume35
dc.identifier.wosWOS:000585969100014
dc.identifier.wosqualityN/A
dc.indekslendigikaynakWeb of Science
dc.language.isoen
dc.publisherUniv Nis
dc.relation.ispartofFacta Universitatis-Series Mathematics and Informatics
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı
dc.rightsinfo:eu-repo/semantics/closedAccess
dc.snmzKA_20241229
dc.subjectTauberian conditions; weight function; summable integrals; finite interval
dc.titleTAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS
dc.typeArticle

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