Representation of Fourier integrals as sums. I

Representation of Fourier Integrals as Sums. Iip

provided an = sin (wn/2) and k(x) =sin (irx/2). Weinberger [8] treated the case when the coefficients a„ form a periodic sequence; that is an+q = a„ for some integer q. Making use of the character theory of Dirichlet 7-functions, he found the conditions on the a„ so that (3) holds with the kernel (2/q112) cos (2irx/q) or the kernel (2/q112) sin (2irx/q). (These kernels give self-reciprocal tran...

Representation of Fourier Integrals as Sums

Here sn x is an abbreviation for sin (wx/2). This paper gives other conditions for the validity of these identities. The previous conditions permitted to have various types of discontinuity. The present paper is concerned with smooth functions; however, the growth at 0 and «> is permitted to be greater than before. Theorems 1, 2, and 3 of the previous paper together with Theorems 2 and 3 of...

Stochastic Processes as Fourier Integrals and Dilation of Vector Measures

Rapidly Growing Fourier Integrals

1. THE RIEMANN–LEBESGUE LEMMA. In its usual form, the Riemann– Lebesgue Lemma reads as follows: If f ∈ L1 and f̂ (s) = ∫∞ −∞ eisx f (x) dx is its Fourier transform, then f̂ (s) exists and is finite for each s ∈ R and f̂ (s) → 0 as |s| → ∞ (s ∈ R). This result encompasses Fourier sine and cosine transforms as well as Fourier series coefficients for functions periodic on finite intervals. When the i...

Surprising Sinc Sums and Integrals

We show that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals. 1 Motivation Recall that sinc(x) := sin(x)/x when x 6= 0 and sinc(0) := 1. In [4] and [8], it was shown that, for N = 0, 1, 2, 3, 4, 5, and 6, ∫ ∞