Analyticity and uniform stability of the inverse singular Sturm--Liouville spectral problem

We prove that the potential of a Sturm--Liouville operator depends analytically and Lipschitz continuously on the spectral data (two spectra or one spectrum and the corresponding norming constants). We treat the class of operators with real-valued distributional potentials in the Sobolev class W^{s-1}_2(0,1), s\in[0,1].Comment: 25 page

On Basicity of the Perturbed System of Exponents in Morrey-Lebesgue Space

In this work, the double system of exponents with complex-valued coefficients is considered. Special cases of these systems often arise when solving problems of mathematical physics by Fourier method. A lot of research by Paley-Wiener, N. Levinson and others have been dedicated to the basis properties of these systems. We find conditions on the coefficients of the system which guarantee its basicity in Morrey-Lebesgue type spaces

On the basicity of unitary system of exponents in the variable exponent Lebesgue spaces

In the present work it is considered the system of functions a (t) e int −b (t) e −int , n ∈ N, with complex-valued coefficients a (·) ; b (·) : [0, π] → C. Sufficient conditions on the coefficients of the system are found in order for the system to form a basis in Lebesgue spaces with variable exponent

Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A

On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and exponentially fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyze and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and \pi. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation.Comment: Revised versio

Wavelets in Statistics

In this paper we give the main uses of wavelets in statistics, with emphasis in time series analysis. We include the fundamental work on non parametric regression, which motivated the development of techniques used in the estimation of the spectral density of stationary processes and of the evolutionary spectrum of locally stationary processes