Fourier Expansions

Multivariate modified Fourier expansions

In this paper, we review recent advances in the approximation of multivariate functions using eigenfunctions of the Laplace operator subject to homogeneous Neumann boundary conditions. Such eigenfunctions are known explicitly on a variety of domains, including the d-variate cube, equilateral triangle and numerous other higher dimensional simplices. Practical construction of truncated expansions...

Fourier Expansions of Arithmetical Functions

On the Fourier expansions of Jacobi forms

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the form pq or p2, there are ...

Fourier - Hermite Expansions for Nonlinear Filtering

The objective of this paper is to develop an approach to nonlinear ltering which allows the separation of time consuming computations involving the coe cients of the system from those dealing with the observation data only. The approach is based on a Cameron-Martin type expansion for nonnormalized optimal lters. 2

The Geometry of Planar Fourier Expansions

Let A be an appropriate planar domain and let f be a piecewise smooth function on R2. We discuss the rate of convergence of S f(x) = Z A b f( ) exp(2 i x)d in terms of the interaction between the geometry of A and the geometry of the singularities of f . The most subtle case is when x belongs to the singular set of f and here Hilbert transform techniques play an important role. The pointwise co...