Abstract

. Using Fourier transform techniques, we establish inequalities for integrals of the form Z 1 0 n Y k=1 sin(a k x) x dx: We then give quite striking closed form evaluations of such integrals and nish by discussing various extensions and applications. 1. Introduction. Motivated by questions about the integral 1 (1) := Z 1 0 1 Y k=1 cos x k dx; we study the behaviour of integrals of the form Z 1 0 n Y k=1 sin(a k x) x dx: In Section 2 we use Fourier transform theory to establish monotonicity properties of these integrals as functions of the parameters. In Section 3, by direct methods, we give closed forms for these integrals and for similar integrals also incorporating cosine terms. In Section 4, we provide a very dierent proof of one of these results following an idea in an 1885 paper of Stormer [2]. Finally, in Section 5 we return to the study of (1). 2. Fourier cosine transforms and sinc integrals. Dene sinc(x) := sin x x ; This research was supported i...

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