In a recent paper, M. B. Nathanson [Nathanson] finds the highest order term of the asymptotic formula for the number of partitions with parts in a finite set. This is done without referring to the partial fraction expansion of the generating function. In this paper, we essentially find the entire asymptotic formula without using partial fractions. This is done by the “metaphysical method” descr...

We show that the multiple zeta sum: ζ(s1, s2, ..., sd) = ∑ n1>n2>...>nd 1 n1 1 n s2 2 ...n sd d , for positive integers si with s1 > 1, can always be written as a finite sum of products of rapidly convergent series. Perhaps surprisingly, one may develop fast summation algorithms of such efficiency that the overall complexity can be brought down essentially to that of one-dimensional summation. ...

THE HYBRID ARCHITECTURE PARALLEL FAST FOURIER

at the target knots yj ∈ [−14 , 1 4 ], j = 1, . . . ,M , where σ = a + ib, a > 0, b ∈ R denotes a complex parameter. Fast Gauss transforms for real parameters σ were developed, e.g., in [15, 8, 9]. In [12], we have specified a more general fast summation algorithm for the Gaussian kernel. Recently, a fast Gauss transform for complex parameters σ with arithmetic complexity O(N log N + M) was int...

Abstract The Fast Fourier Transform (FFT) is a cornerstone of digital signal processing, generating computationally efficient estimate the frequency content time series. Its limitations include: (1) information only provided at discrete steps, so further calculation, for example interpolation, often used to obtain improved estimates peak frequencies and amplitudes; (2) ‘energy' from sp...