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The Parseval formulae for monotonic functions. II

Published online by Cambridge University Press:  24 October 2008

Sheila M. Edmonds
Sheila M. Edmonds
Affiliation:
Newnham CollegeCambridge
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Extract

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 2 , April 1950 , pp. 231 - 248
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

† ‘The Parseval formulae for monotonic functions. I’, Proc. Cambridge Phil. Soc. 43 (1947), 289–306Google Scholar. This paper will be referred to in future as M.F. (I).

‡ ‘On the Parseval formulae for Fourier transforms’, Proc. Cambridge Phil. Soc. 38 (1942), 1–19Google Scholar. This paper will be referred to as P.F.

§ ‘The Parseval formulae for monotonic functions. III’, Proc. Cambridge Phil. Soc. 46 (1950), 249–267Google Scholar. This paper will be referred to as M.F. (III).

∥ I take this opportunity of mentioning a correction needed in P.F.; the minus sign on the right-hand side of this formula was accidentally omitted there. This slip did not affect the working in P.F. since none of the proofs dealt with the ‘complex’ formula.

† ‘The Parseval formulae for monotonic functions. IV’ (to be published). In M.F. (I) a reference to my dissertation was given for this point, as I did not at that time intend to publish the proof.

‡ There is also the trivial complication that the sum of a sine series with decreasing coefficients b n is ‘not quite positive’ in (O, π); we can only show that it is not less than For the purposes of the above discussion such series should be classed with the ‘positive’ cases.

† We note that in cases [A, 2], [B, 2], [C, 1] and [C, 2], the series (2) in fact consists entirely of cosine coefficients or entirely of sine coefficients (since at least one of f, g is defined by a cosine or sine series); thus (2) reduces to a form similar to (3). It is easy to pick out wucli oases in the tables as the cosine and sine results appear separately.

† This system is slightly complicated by the fact that, for later convenience, we have interchanged f and g in passing from the transform to the series theorem. Thus f, G c in [C] correspond to g, a n in [C, 1] for instance.

† The method is really a disguised form of a technique developed by Young, (Proc. Roy. Soc. A, 85 (1911), 401–14)CrossRefGoogle Scholar, and now familiar. It is first proved that, if f and g are any periodic integrable functions, then the function has the Fourier coefficients π(αn αn + b nβn), π(αnb n − a n βn). We have then only to consider the convergence of the Fourier series of h at the origin; in the present case, this is ensured by the Dirichlet-Jordan test, since the methods used above show that h is a continuous function of bounded variation. We could thus have omitted the working down to (4) by quoting Young's result; but we have preferred to follow the lines of the transform proof.

† Zygmund, A., Trigonometrical series (Warsaw, 1935), p. 91.Google Scholar

‡ Compare, Hobson E. W., The theory of functions of a real variable, 2 (Cambridge, 1926), p. 591.Google Scholar

† The integral involving f * and g * does not come under Lemma 7, but is obviously a bounded increasing function of y.

† To prove this, we have only to substitute for with a similar substitution for and then integrate from 0 to 2π.

‡ There is, of course, a corresponding simplification of the transform proof in the sine case of Theorem 1; this proof is even easier than for series, and we give it here to serve as a model for the series proof, since the latter is only outlined above.

† See M.F.(I), pp. 301, 302.