Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients

Modified Fourier expansions: theory, construction and applications

Modified Fourier expansions present an alternative to more standard algorithms for the approximation of nonperiodic functions in bounded domains. This thesis addresses the theory of such expansions, their effective construction and computation, and their application to the numerical solution of partial differential equations. As the name indicates, modified Fourier expansions are closely related to classical Fourier series. The latter are naturally defined in the d-variate cube, and, in an analogous fashion, we primarily study modified Fourier expansions in this domain. However, whilst Fourier coefficients are commonly computed with the Fast Fourier Transform (FFT), we use modern numerical quadratures instead. In contrast to the FFT, such schemes are adaptive, leading to great potential savings in computational cost. Standard algorithms for the approximation of nonperiodic functions in $d$-variate cubes exhibit complexities that grow exponentially with dimension. The aforementioned quadratures permit the design of approximations based on modified Fourier expansions that do not possess this feature. Consequently, such schemes are increasingly effective in higher dimensions. When applied to the numerical solution of boundary value problems, such savings in computational cost impart benefits over more commonly used polynomial-based methods. Moreover, regardless of the dimensionality of the problem, modified Fourier methods lead to well-conditioned matrices and corresponding linear systems that can be solved cheaply with standard iterative techniques. The theoretical component of this thesis furnishes modified Fourier expansions with a convergence analysis in arbitrary dimensions. In particular, we prove uniform convergence of modified Fourier expansions under rather general conditions. Furthermore, it is known that the notion of modified Fourier expansions can be effectively generalised, resulting in a family of approximation bases sharing many of the features of the modified Fourier case. The purpose of such a generalisation is to obtain both faster rates and higher degrees of convergence. Having detailed the approximation-theoretic properties of modified Fourier expansions, we extend this analysis to the general case and thereby verify this improvement. A central drawback of these expansions is that their convergence rate is both fixed and typically slow. This makes the construction of effective convergence acceleration techniques imperative. In the final part of this thesis, we design and analyse a robust method, applicable in arbitrary numbers of dimensions, for accelerating convergence of modified Fourier expansions. When employed in the approximation of multivariate functions, this culminates in efficient, high-order approximants comprising relatively small numbers of terms

Perturbed eigenvalues of polyharmonic operators in domains with small holes

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy-Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary

On the spectral properties of the differential operators with involution

In this paper we deal with the problems of the eigenfunction expansions related to the differential operators with involution. The mean value formula for the eigenfunction is obtained with application of the transformation methods of the operators in the symmetric regions. The obtained formula is applied to estimate the eigenfunctions of the given differential operator in the ball. For domains with smooth boundary, the solution to these differential operator problems involves eigenfunction expansions associated with biharmonic-type operator with Navier boundary conditions