In this paper we are concerned with the inverse spectral problems for energy-dependent Sturm-Liouville problems (that is, differential pencils) defined on interval [0, 1] with two potentials known on a subinterval [a1, a2] ⊂ [0, 1]. We prove that the potentials on the entire interval [0, 1] and the boundary condition at x = 1 are uniquely determined in terms of partial knowledge of the spectrum...

In the study of stationary stochastic processes on the real line, the covariance function and the spectral density function are parameters of considerable interest. They are equivalent ways of expressing the temporal dependence in the process. In this article, we consider the spectral density function and propose a new estimator that is not based on the periodogram; the estimator is derived thr...

We present a new method to solve an inverse scattering problem for time-harmonic electromagnetic waves. The method is based on an equivalence relation of the inverse problem to a system of two integral equations. This system can be solved numerically by a recently developed spectral method for the direct scattering problem.

Spectral analysis considers the problem of determining (the art of recovering) the spectral content (i.e., the distribution of power over frequency) of a stationary time series from a finite set of measurements, by means of either nonparametric or parametric techniques. This paper introduces the spectral analysis problem, motivates the definition of power spectral density functions, and reviews...

In the first edition of this book the main attention was focused on the methods of solving the inverse problem of spectral analysis and on the conditions (necessary and sufficient) which the spectral data must satisfy in order to make it possible to reconstruct the potential of the corresponding Sturm-Liouville operator. These conditions imply that the spectral data (e.g. spectral function or s...

‎in this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎the equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

In this paper, we deal with the inverse spectral problem for the equation −(pu′)′ + qu = λρu on a finite interval (0, h). We give some uniqueness results on q and ρ from the Gelfand spectral data, when the coefficients p and ρ are piecewise Lipschitz and q is bounded. We also prove an equivalence result between the Gelfand spectral data and the Borg–Levinson spectral data. As a consequence, we ...

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