(R1521) On Weighted Lacunary Interpolation

In this paper, we considered the non-uniformly distributed zeros on the unit circle, which are obtained by projecting vertically the zeros of the derivative of Legendre polynomial together with x=1 and x=-1 onto the unit circle. We prescribed the function on the above said nodes, while its second derivative at all nodes except at x=1 and x=-1 with suitable weight function and obtained the existence, explicit forms and establish a convergence theorem for such interpolatory polynomial. We call such interpolation as weighted Lacunary interpolation on the unit circle

On the \u3ci\u3ep\u3c/i\u3e-Inner Functions of ℓ\u3csup\u3ep\u3c/sup\u3e\u3csub\u3eA\u3c/sub\u3e

Define ℓpA as the space of all functions holomorphic over the unit disk whose Taylor coefficients are p-summable. Despite their classical origins and simple definition, these spaces are not as well understood as one might expect. This is particularly true when compared with the Hardy spaces, which provide a useful road map for the types of questions we might consider reasonable. In this work we examine the zero sets of ℓpA, p ∈ (1;∞), as well as a notion of inner function that is consistent with the approach taken on numerous other function spaces. Basic properties of p-inner functions are proved. It is shown that for some values of p, there are Blaschke sequences that fail to be a zero set for ℓpA. It is also shown that canonical factorization fails for ℓpA

Fine Structure of the Zeros of Orthogonal Polynomials: A Review

We review recent work on zeros of orthogonal polynomials

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

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given