Almost Periodic Functions by Harald Bohr

Author:Harald Bohr
Language: eng
Format: epub, pdf
Publisher: Dover Publications, Inc.
Published: 2018-12-14T16:00:00+00:00

holds. The proof of this equivalence is completely analogous to the corresponding proof for periodic functions.

1. The uniqueness theorem follows from Parseval’s equation.

Proof; Let f(x) be an almost periodic function, whose Fourier series has no terms whatsoever. Then from Parseval’s equation M{|f(x)|2} = Σ|An|2, it follows that M{|f(x)|2} = 0. Our problem is to show on the basis of M{|f(x) |2} = 0 that f(x) vanishes identically. For purely periodic functions the corresponding conclusion is trivial, since, in fact, the mean value of a non negative continuous function over a finite interval can vanish only if the function vanishes identically. For almost periodic functions, where we are taking a mean value over an infinite interval the conclusion is (fortunately) still true, although not immediate. We shall defer the proof to the next paragraph in order to prove for later use some further theorems in the same connection.

2. Parseval’s theorem follows from the uniqueness theorem.

Proof: Let be an arbitrary almost periodic function. We form the following function (which, by formula (7) in § 66 is itself almost periodic) :

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