By M. H. Taibleson

This ebook offers a improvement of the elemental proof approximately harmonic research on neighborhood fields and the *n*-dimensional vector areas over those fields. It focuses nearly solely at the analogy among the neighborhood box and Euclidean situations, with appreciate to the shape of statements, the style of evidence, and the range of applications.

The strength of the analogy among the neighborhood box and Euclidean situations rests within the dating of the sector constructions that underlie the respective circumstances. a whole category of in the neighborhood compact, non-discrete fields offers us examples of hooked up fields (real and complicated numbers); the remainder are neighborhood fields (p-adic numbers, p-series fields, and their algebraic extensions). The neighborhood fields are studied with a view to expand wisdom of the reals and complexes as in the neighborhood compact fields.

The author's vital goal has been to give the elemental evidence of Fourier research on neighborhood fields in an available shape and within the comparable spirit as in Zygmund's *Trigonometric Series* (Cambridge, 1968) and in *Introduction to Fourier research on Euclidean Spaces* by means of Stein and Weiss (1971).

Originally released in 1975.

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