Spectral parameter power series representation for Hill's discriminant

We establish a series representation of the Hill discriminant based on the spectral parameter power series (SPPS) recently introduced by V. Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and employing the Mathieu case the feasibility of this type of series for numerical calculations of the eigenspectrumComment: 13 pages, 2 figures, 19 references with titles, a few minor changes in the tex

On the inversion of integral transforms associated with Sturm-Liouville problems

AbstractConsider the Sturm-Liouville boundary-value problem 1.(1) y″ − q(x) y = −t2y, −∞ < a ⩽ x ⩽ b < ∞2.(2) y(a) cos α + y′(a) sin α = 03.(3) y(b) cos β + y′(b) sin β = 0, where q(x) is continuous on [a, b]. Let φ(x, t) be a solution of either the initial-value problem (1) and (2) or (1) and (3). In this paper we develop two techniques to invert the integral F(t) = ∝abf(x) φ(x, t) dx, where f(x) ϵ L2(a, b); one technique is based on the construction of some biorthogonal sequence of functions and the other is based on Poisson's summation formula

Generalizing sampling theory for time-varying Nyquist rates using self-adjoint extensions of symmetric operators with deficiency indices (1,1) in Hilbert spaces

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions

Eigenvalue problems, spectral parameter power series, and modern applications

"Our review is dedicated to a wide class of spectral and transmission problems arising in di?erent branches of applied physics. One of the main di?culties in studying and solving eigenvalue problems for operators with variable coe?cients consists in obtaining a corresponding dispersion relation or characteristic equa-tion of the problem in a su?ciently explicit form. Solutions of the dispersion relation are the eigenvalues of the problem. When the dispersion relation is known the eigenvalues are found numerically even for relatively simple problems with constant coe?cients because even in those cases as a rule the dispersion relation represents a transcendental equation the exact solutions of which are unknown.

Spectral theory using linear systems and sampling from the spectrum of Hill's equation

This thesis, entitled Spectral Theory Using Linear Systems and Sampling from the Spectrum of Hill’s Equation is submitted by Caroline Brett, Master of Science for the degree of Doctor of Philosophy, September 2015. It uses linear systems to solve various problems connected with Hill’s equation, −f + qf = λf for q ∈ C2, real-valued and π-periodic. Introducing a new operator, Rx constructed from a linear system, (−A, B, C) allows us to solve Hill’s equation and the inverse spectral problem. We use Rx to construct a function, T(x, y) that satisfies a Gelfand–Levitan integral equation and then derive a PDE for T(x, y). Solving this PDE recovers q. Extending Hill’s work in [28], we show that there exist Hilbert–Schmidt operators, Rp and Rc analogous to Rx, such that the roots of their Carleman determinants are elements of the periodic spectrum of Hill’s equation. The latter half concerns sampling from entire functions in Paley–Wiener space. From the periodic spectrum of Hill’s equation we derive a sampling sequence, (tn)n∈Z. Whittaker, Kotel’nikov and Shannon give a sampling result for (n)n∈Z where samples occur at a constant rate. Samples taken from the periodic spectrum do not occur at a constant rate, nevertheless we provide analogous results for this case. From (tn)n∈Z we also construct Riesz bases for L2[0, π] and L2[−π, π], the Fourier transform space of PW(π). In L2[0, π] we construct the dual Riesz basis using linear systems. Furthermore, we show that the determinant of the Gram matrix associated with the Riesz basis is a Lipschitz continuous function of (tn)n∈Z. Finally, we look at an integral, Ia associated with Ramanujan and use it to create a basis for PW π2. We conclude with an evaluation of various determinants associated with Ia

The inverse problem for a spectral asymmetry function of the Schrodinger operator on a finite interval

For the Schroedinger equation $−d^2u/dx^2 + q(x)u = λu$ on a finite $x$-interval, there is defined an “asymmetry function” $a(λ; q)$, which is entire of order 1/2 and type 1 in $λ$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function $a(λ)$. For any given $a(λ)$, there is one potential for each Dirichlet spectral sequence

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations