SOME ASPECTS OF NORLUND SUMMABILITY METHODS

Abstract

The dissertation consists of six chapters. The first chapter is a collection of well-known basic results concerning inclusion and equivalence of Norlund methods, with some gaps filled in. In Chapter II, the relation between Norlund and Casaro methods are discussed; some remarks on the proofs of theorems concerning Norlund methods which include Casaro methods of all positive orders are given here. The following chapter contains mostly new results which show that the regularity of (N,pn) and (N,qn) implies the regularity of ( N, Pn +iqn), but that the converse is not true. A brief survey on the problem of consistency and the relation with Abel means is made in Chapter IV and some comments are made at the end of the chapter. Chapter V deals with the total inclusion problem in which a remark is made on the total inclusion relations between (R,n,k) and ((C,k). The last chapter is a short account of the fourier-effectiveness of a certain class of Norlund methods, in which we obtain a generalization of a particular Norlund method. The proofs of lemma 2.4.3+, Theorem 2.4.1 and theorem 4.1.5, which are not available explicitly in the literature, are presented here. Theorem 6.2.2 and Theorem 6.2.3 are the generalization of the existing theorems. The following are simple results deduced from basic theorems : Corollary 1.2.3, Corollary 1.2.4, Corollary 1.2.5, Corollary 1.3.1, Corollary 1.3.2, Corollary 1.3.4, Corollary 1.3.5, Corollary 1.4.2, Theorem 2.2.4, Theorem 2.2.5 Theorem 3.1.1, Corollary 3..1, Lemma 3.1.3. The new results, (of which the more important ones are marked the asterisks) are as follows: *Theorem 1.3.3, Theorem 1.3.7, Lemma 1.5.2 *Theorem 1.5.2, Theorem 1.6.9, *Theorem 3.1.2, Corollary 3.1.2, *Theorem 3.1.3, Lemma 3.1.1, *Theorem 3.1.4, Theorem 3.1.5, Theorem 3.1.6, Theorem 3.1.7, Theorem 3.1.8, Theorem 3.1.9, *Theorem 3.1.10, Corollary 3.1.6, Lemma 3.1.4 *Lemma 3.1.5, Theorem 3.1.11, *Theorem 3.2.1, *Theorem 5.1.13. Lastly, comments on the results and their profits have been included in the various remarks inserted at different places in the dissertation.