Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Let r > 0, (alpha) > 0, (alpha)N(,o) + (beta) > 0 where N(,o) is a non-negative integer, and let;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);The Borel-type summability method (B,(alpha),(beta)) is defined as follows: s(,n) (--->) L(B,(alpha),(beta)) or;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);The special case (B,1,1) is the ordinary Borel exponential method B.;The main result of this thesis is the following Tauberian theorem for summability (B,(alpha),(beta)):;If s(,n) is a real sequence that satisfies the slowly decreasing condition;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);and s(,n) (--->) L(B,(alpha),(beta)), then s(,n) (--->) L(C(,2r)). Here C(,2r) denotes the Cesaro method of order 2r.;In particular, if a(,n) = O(,L)(n('r- 1/2)), n = 1,2,..., then (T(,r)) holds and thus the result extends a theorem due to Kwee.;It is shown, by a Vijayaraghavan type theorem, that (T(,r)) and s(,n) (--->) L(B,(alpha),(beta)) together imply that s(,n) = O(n('r)). Next a known method of summability V(,(alpha)) is considered. This is a special case of the Valiron method and it is shown that summability (B,(alpha),(beta)) is equivalent to summability V(,(alpha)) provided that s(,n) = O(n('r)). The desired Tauberian theorem is then deduced from established results.



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