Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

Solutions to Some Real-Life Problems Based on Mathematical Modeling and Functional Minimization

Building mathematical models that can describe, predict, and explain real-life phenomena is useful. This paper features the functional dependency model and the square of this functional dependency which hold significant information. A mathematical model that relates these functional dependencies with the average value of the function was developed to show that for an arbitrary well-behaved function, the definite integral of the square of the function over a finite interval is minimal when the function is constant over the interval. Finally, the model’s validity and accuracy in representing real-world problems for different situations in physics like mechanics, quantum mechanics, and electricity in economics were evaluated.

A 64 bit quantum dragon data-set for machine learning

Design and examples of a sixty-four bit quantum dragon data-set are presented. A quantum dragon is a tight-binding model for a strongly disordered nanodevice, but when connected to appropriate semi-infinite leads has complete electron transmission for a finite interval of energies. The labeled data-set contains records which are quantum dragons, which are not quantum dragons, and which are indeterminate. The quantum dragon data-set is designed to be difficult for trained humans and machines to label a nanodevice with regard to its quantum dragon property. The 64 bit record length allows the data-set to be utilized in restricted Boltzmann machines which fit well onto the D-Wave 2000Q quantum annealer architecture.

Hyers–Ulam stability of impulsive Volterra delay integro-differential equations

This paper discusses different types of Ulam stability of first-order nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory effects represented by the delay and the kernel of the used fractional integral operator. Our analysis is based on Pachpatte’s inequality and the fixed point approach represented by the Picard operators. Applications are provided to illustrate the stability results obtained in the case of a finite interval.

Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

Propagation of Electrical Pulse in the Finite Interval of Perfect Electrical Conductivity and its Full Absorption at the End of Interval

The propagation of an electric pulse in a finite interval is investigated in the case when the pulse is generated at the input of the interval and absorbed at the end of the interval. The pulse propagation is described by a hyperbolic equation with regard for dissipation. The pulse generation at the input is specified as a Heaviside function, and the absorption at the output is set by a permanent magnet. The model describes the propagation of disturbances with a finite speed. A formulation of the corresponding initial boundary value problem is given, for the solution of which the Laplace transform in time is applied in the case of arbitrary coefficients. An exact analytical solution in the Laplace image space was obtained, and other applications with the complete absorption are presented. A general solution is constructed, and the case of low dissipation is considered for some values of the coefficients characterizing real situations.