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Smooth Global Approximation For Continuous Data Assimilation, Kenneth R. Brown

Theses and Dissertations

This thesis develops the finite element method, constructs local approximation operators, and bounds their error. Global approximation operators are then constructed with a partition of unity. Finally, an application of these operators to data assimilation of the two-dimensional Navier-Stokes equations is presented, showing convergence of an algorithm in all Sobolev topologies.

Interpolation And Sampling In Analytic Tent Spaces, Caleb Parks

Graduate Theses and Dissertations

Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in harmonic analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Mobius invariance for the tent spaces.

Modified Pal Interpolation And Sampling Bilevel Signals With Finite Rate Of Innovation, Gayatri Ramesh

Electronic Theses and Dissertations

Sampling and interpolation are two important topics in signal processing. Signal processing is a vast field of study that deals with analysis and operations of signals such as sounds, images, sensor data, telecommunications and so on. It also utilizes many mathematical theories such as approximation theory, analysis and wavelets. This dissertation is divided into two chapters: Modified Pal´ Interpolation and Sampling Bilevel Signals with Finite Rate of Innovation. In the first chapter, we introduce a new interpolation process, the modified Pal interpolation, based on papers by P ´ al, J ´ oo´ and Szabo, and we establish the existence and …

Minimal And Symmetric Global Partition Polynomials For Reproducing Kernel Elements, Mario Jesus Juha

USF Tampa Graduate Theses and Dissertations

The Reproducing Kernel Element Method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker-δ property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This dissertation shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use in previous formulations of incomplete polynomials that are subsequently non-affine invariant. This dissertation lays out …

Lifting Module Maps Between Different Noncommutative Domain Algebras, Jonathan Von Stroh

Electronic Theses and Dissertations

The classical Carathéodory interpolation problem is the following: let n be a natural number, a₀, a₁, . . . , a_N be complex numbers, and D the unit disk. When does there exist an analytic function F : D → C and complex numbers a_N+1, a_N+2, . . . such that F(z) = a₀ + a₁z + a₂z² + . . . + a_Nz^N + a_N+1z^N+1 + . . . and ||F||_∞ < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.

Arias presented generalizations of the original …

Fractal Interpolation, Gayatri Ramesh

Electronic Theses and Dissertations

This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton s, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley s paper, Fractal Functions and Interpolation. I also mention results on iterated function system …

Rational Cubic B-Spline Interpolation And Its Applications In Computer Aided Geometric Design, Kotien Wu

Mathematics & Statistics Theses & Dissertations

Because of the flexibility that the weights and the control points provide, NURBS have recently become very popular tools for the design of curves and surfaces. If the weights are positive then the NURB will lie in the convex hull of its control points and will not possess singularities. Thus it is desirable to have positive weights.

In utilizing a NURB a designer may desire that it pass through a set of data points {x_i} This interpolation problem is solved by the assigning of weights to each data point. Up to now little has been known regarding the …