In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: – The first negative coefficient of the sequence {af(tn)}n∈N, – The number of coefficients af(tn ) of same signs, – Non-vanishing of coefficients af(tn ) in short intervals and in arithmetic progressions, where af(n) is the n-th Fourier coefficient of f and t is a square...

We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize coefficients introduced by Bump and Ginzburg for the double cover. The covers for which these coefficients vanish identically (resp. do not vanish for some ...

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f(z) = P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If ` is prime, then we shall be interested in congruences of the form

In this paper we prove a basic theorem which says that if f : Fpn → [0, 1] has few ‘large’ Fourier coefficients, and if the sum of the norms squared of the ‘small’ Fourier coefficients is ‘small’ (i.e. the L2 mass of f̂ is not concetrated on small Fourier coefficients), then Σn,df(n)f(n+ d)f(n + 2d) is ‘large’. If f is the indicator function for some set S, then this would be saying that the set...

We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378–391] and we extend their results. Construction of new Gaussian quadrature form...

When Fourier techniques are employed to specific option pricing cases from computational finance with non-smooth functions, the so-called Gibbs phenomenon may become apparent. This seriously impacts the efficiency and accuracy of the pricing. For example, the Variance Gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier se...

Fourier transform is among the most widely used tools in computer science. Computing the Fourier transform of a signal of length N may be done in time Θ(N logN) using the Fast Fourier Transform (FFT) algorithm. This time bound clearly cannot be improved below Θ(N), because the output itself is of length N . Nonetheless, it turns out that in many applications it suffices to find only the signifi...

Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigenvalues of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associa...