SpringerBriefs in Mathematics / An Introduction to Ultrametric Summability Theory

Preface.- Introduction and Preliminaries.- Some Arithmetic and Analysis in Qp : Derivatives in Ultrametric Analysis.- Ultrametric Functional Analysis.- Ultrametric Summability Theory.- References.- Index.

Pinnangudi Narayanasubramanian Natarajan is former professor and head, Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai. He did his PhD from the University of Madras, under Prof. M.S. Rangachari, former director and head, The Ramanujan Institute for Advanced Study in Mathematics, University of Madras. An active researcher, Prof. Natarajan has over 90 research papers to his credit published in several international journals like Proceedings of the American Mathematical Society, Bulletin of the London Mathematical Society, Indagationes Mathematicae, Annales Mathematiques Blaise Pascal, Commentationes Mathematicae (Prace Matematyczne). His research interest includes summability theory and functional analysis (both classical and ultrametric). Prof. Natarajan was honored with Dr. Radhakrishnan Award for the Best Teacher in Mathematics for the year 1990-91 by the Government of Tamil Nadu. Besides visiting several institutes of repute in Canada, France, Holland, and Greece on invitation, Prof. Natarajan has participated in several international conferences and chaired sessions, too.

Most of the material discussed in the book An Introduction to Ultrametric Summability Theory has not appeared in a book form before. The Silverman Toeplitz theorem in an ultra-metric field turns into the following form: an infinite matrix or a summability matrix A = ( a nk ) preserves convergent sequences if and only if sup n ; k a nk < Yen ; lim n ® Yen a nk = d k exists, k = 0, 1, 2, and lim n ® Yen exists. The theorem is proved by using the sliding-hump method avoiding traditional tools like the signum function. Schur s theorem, Steinhaus theorem, and Knopp s core theorem are discussed. It is proved that neither certain Steinhaus type theorems nor the Mazur Orlicz theorem and Brudno s theorem holds. Certain special summability methods, such as the Norlund method, weighted mean method, Euler method, Taylor method and their properties are extensively discussed. Some product theorems and Tauberian theorems are also proved. Double sequences and double series are discussed, and the Silverman Toeplitz theorem for four-dimensional infinite matrices is given. Some applications of this theorem are also discussed in the book. ( Hüseyin Çakalli, Maltepe University, Istanbul, Turkey ) - See more at: