We consider the use of eigenfunctions of polyharmonic operators, equipped with homogeneous Neumann boundary conditions, to approximate nonperiodic functions in compact intervals. Such expansions feature a number of advantages in comparison with classical Fourier series, including uniform convergence and more rapid decay of expansion coefficients. Having derived an asymptotic formula for expansi...

Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite functions. They arise from a generalization the expansion eigenstates Laplacian in regular polygons. A simple accurate approximation J0(x) is found on interval [0, 2].

Achieving improved order of accuracy for a hydrodynamic numerical method is a continuing quest. The discrete approximate solution error, in general, can be expressed as a truncation of a Taylor series expansion. This paper presents a weak statement perturbation methodology retaining block-tridiagonal forms that can reduce, or annihilate in special cases, the Taylor series truncation error to hi...

A quasi-interpolation formula based on discrete function values is given in the form of a multivariate spline series that yields the local approximation order characterized by the Fix-Strang conditions. This formula can be considered as a partial sum of the formal Neumann series expansion of the formal interpolation operator, and hence, justifies that "quasi-interpolation" is indeed an appropri...

A simple and efficient FFT-based fast direct solver for Poisson-type equations on 3D cylindrical and spherical geometries is presented. The solver relies on the truncated Fourier series expansion, where the differential equations of Fourier coefficients are solved using second-order finite difference discretizations without pole conditions. Three different boundary conditions (Dirichlet, Neuman...

Aims. In view of the substantial uncertainties regarding the possible dynamics of the dark energy, we aim at constraining the expansion rate of the universe without reference to a specific Friedmann model and its parameters. Methods. We show that cosmological observables integrating over the cosmic expansion rate can be converted into a Volterra integral equation which is known to have a unique...

We consider second order linear differential equations in a real interval I with mixed Dirichlet and Neumann boundary data. We consider a representation of its solution by a multi-point Taylor expansion. The number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent ∀ x ∈ I. We propose several algorithms to approximat...

There is a class of single trace operators in N = 4 Yang-Mills theory which are related by the AdS/CFT correspondence to classical string solutions. Interesting examples of such solutions corresponding to periodic trajectories of the Neumann system were studied recently. In our paper we study a generalization of these solutions. We consider strings moving with large velocities. We show that the...