Volume 3, Issue 2

Abstract. In this work, by using the Dunkl transform operator, we extend the Donoho–Stark uncertainty principle to this transform. A special interest is devoted to the computation of the spectrum of the finite Dunkl transform. For this purpose, we develop two different practical methods. Also, we show how the eigenfunctions of the finite Dunkl transform can be used to reconstruct in a stable way the Dunkl bandlimited signals. Finally, we present some numerical examples that illustrate the results of this work.

Abstract. In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem in , where is an annulus in , , and satisfies some appropriate conditions. We establish a priori estimates of radial solutions with prescribed number of zeros. Moreover, when , where is odd and super-linear at infinity, using the uniqueness result due to Tanaka (2007), we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly .

Abstract. We construct a new solution-generating mapping between nonlinear, ordinary differential equations of second order, based on a combination of Sundman and Darboux transformations.

Abstract. The concepts of multiresolution analysis (MRA) and wavelet can be generalized to a local field of positive characteristic by using a prime element of such a field. An MRA is a sequence of closed subspaces of satisfying certain properties. We show that it is enough to assume that the discrete translates of a single function in the core subspace of the MRA form a Riesz basis instead of an orthonormal basis and show how to construct an orthonormal basis from a Riesz basis. We also prove that the intersection triviality condition in the definition of MRA follows from the other conditions of an MRA. The union density condition also follows if we assume that the Fourier transform of the scaling function is continuous at 0. Finally we characterize the scaling functions associated with such an MRA.

Abstract. This article shows the existence and multiplicity of positive solutions of the -Laplacian problem where is a bounded open set in with smooth boundary and are continuous functions on such that and are smooth functions which may change sign in . The method is based on Nehari results on three sub-manifolds of the space .

Abstract. Suppose has a Fourier series () with small gaps for all . Here, by applying the Wiener–Ingham result for finite trigonometric sum with `small' gaps, we estimate the order of magnitude of the Fourier coefficients and obtain a sufficient condition for the convergence of the series () if is locally of class .

Abstract. Let be a fixed probability measure. For each dimension , let be i.i.d. -valued radial random variables with radial distribution . We derive two central limit theorems (CLTs) for for with normal limits. The first CLT for follows from known estimates of convergence in the CLT on , while the second CLT for will be a consequence of asymptotic properties of Bessel convolutions. Both limit theorems are considered also for -invariant random walks on the space of matrices instead of for and fixed dimension .