INTERPOLATION AND APPROXIMATION AND CURVE FITTING

Author: PALACIOS QUIÑONERO FRANCISCO.
Year: 2004.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: SALA D'ACTES FME.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Summary: The objective of this thesis is to develop through Birkhoff interpolation polynomials lacunarios. In Birkhoff interpolation algebraic determining a polynomial of degree less than n, this will employ No conditions that set the value of the polynomial or their derivatives. The classic problems of interpolation Lagrange, Taylor, Hermite, Hermite-Sylvester and Abel-Gontcharov are particular cases of algebraic Birkhoff interpolation. An area of polynomials lacunarios dimension n is the set of polynomials that can be generated by linear combination of n powers various degrees, in general, not consecutive. In particular, when we Powers grade 0.1, â | n-1, we obtain the space of polynomials of degree less than n, used in the classical algebraic interpolation. In the classical algebraic interpolation, the number of conditions determines the space interpolation. In contrast, using interpolation polynomials lacunarios conditions interpolation only determine the size of interpolation and there may be an infinite number of spaces on which perform interpolation. This allows us to build better strategies interpolation in certain cases, such as interpolation functions for high growth (exponential interpolation and branches asintóticas). The contribution of the thesis consists in defining a proper theoretical framework for Birkhoff interpolation polynomials lacunarios and by extension to the new framework of the main elements of the algebra of Birkhoff interpolation. Specifically, it was widely Pólya condition is characterized regularity tied down conditions sufficient regularity ordered that extend the theorem Atkhison-Sharma, extending the normal decomposition and establishes sufficient conditions of singularity in cases indescomponibles.

Author: RUBIO MASSEGU JOSEP.
Year: 2004.
University: POLITÉCNICA DE CATALUÑA [www.upc.edu].
Place of defense: Sala d'Actes de l'EUPM.
Place of preparation: U FACULTAT DE MATEMATIQUES I ESTADISTICA SUD.

Author: GODÉS BLANCO CARMEN.
Year: 2005.
University: ZARAGOZA [www.unizar.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: The Geometric Characterization was proposed by Chung and Yao in 1977. This was introduced in order to characterize the sets of nodes space k-dimensional whose problem deinterpolación Lagrange partner is unisolvente and their respective Lagrange polynomials can be expressed as the product of linear factors. The plane, a set of X (n +2) (n + l) / 2 nodes verifies Geometric Characterization of order n (GCn) if, for each xEX There No straight containing all points of X (x ) rather than x. The retículos principal is a classic example of joint GCn. In 1982, Gasca and Maeztu conjeturaron that any set GCn, there is at least a line with n + 1 nodes. So far, this conjecture has only been tested for n = 4. In order to classify sets GCn, Carnicer Gasea introduced, in an article in the year 2001, the concept of default. Specifically, we say that a set GCn has default d if it contains exactly n + 2-d lines of n + 1 nodes. These authors described the joint defects 0.1 and 2 and characterized the joint GC4 faulty 3. It also showed that the default maximum that can be achieved is d = n-1, assuming that the conjecture Gasea and Maeztu is true. With these results addressed the complete description and classification of GCn for n = 4. In this paper, using the assumptions as conjecture Gasea and Maeztu, have been described and classified all sets of nodes of the plane that verify Characterization Geometric GCn for any n. As classification criterion has been used in the default configuration of nodes. Within this context, the first major contribution is the characterization of CCn faulty 3 for n> 4. These profiles contain a subset GC4 faulty 3 which relates to a certain way with the rest of the lines of reticulum. To get the overview of the joint CCn faulty n-1, it has been necessary to introduce a generalization of the retículos major. Our second output interest shows that GCn faulty n-1 equivalent to such retículos major widespread (GPLn). Moreover, it shows that the lines forming a GPLn, are contained in a cubic beam, only if n = 4. Our qualifying concludes with a demonstration of the impossibility of existence sets GCn defects between 4 and n-2 for n> 5. This result has been of vital importance to achieve our goal. Finally, they can also find some numerical experiments, in conjunction with the results. These have been made with the help of a symbolic manipulator.