An Introduction to Ultrametric Summability Theory

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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 = (ank) preserves convergent sequences if and only if supn;k |ank| < [yen]; limn(R)[yen] ank = dk exists, k = 0, 1, 2, ... and limn(R)[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. (Huseyin Cakalli, Maltepe University, Istanbul, Turkey) - See more at: http://www.ams.org/mathscinet/pdf/3136725.pdf?pg1=IID&s1=229805&vfpref=html&r=1