Supervisors info:
Σωτήρης Νοτάρης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Summary:
The current thesis deals with two of the most old and important subjects in the field of
Numerical Analysis and Approximation Theory, Polynomial Interpolation and Numerical
Integration. The first part of our work tries to deal with questions regarding the approximation of a continuous function by an nth degree polynomial, which agrees with f at n + 1 points, known as the interpolating polynomial at these points. In the past, it was widely believed that this approximation method is always successful as the number of interpolating points increases without bound, in the sense that the error of the approximation tends to zero. However, we will present some general results that lead to Faber theorem, which once and for all proves that this is false. In the second part of this section we will show the importance of Chebychev polynomials in polynomial interpolation. More specifically, combining
the properties of the Chebychev nodes of the first kind and Jackson theorem we will be able to show that suppose that f is a Lipschitz function in [−1,1], the error of the interpolation error for the set of Chebychev nodes of the first kind converges to zero.
In the second section, we will shift our attention to Numerical Integration, particularly
to formulas derived from polynomial interpolation, the interpolatory type formulas. After
we state some important properties of interpolatory formulas such us properties regarding
the degree of exactness and the positivity of the weights, we proceed by introducing Gauss formulas, which have optimal degree of exactness among all interpolating formulas. We prove their properties regarding the optimality of their degree of exactness, the positivity of the weights and the error of the formula.
The original work in this thesis is the presentation of a new quadrature formula, which
involves the pairing of the Chebychev weight function of the second kind with Chebychev
nodes of the third kind (called the (2,3) formula) and Chebychev nodes of the fourth kind
(called the (2,4) formula). By relating them to the Chebychev Gauss formula of the third
kind and using the correlation between Chebychev nodes of the third and fourth kind, we
are able to find the weights and the degree of exactness of the formula. In addition, using a functional analysis technique, by viewing the error as a functional, under specific hypotheses, one can compute the norm of the error functional. Proving that such conditions are satisfied in our case, we explicitly compute the error norm of the (2,3) and (2,4) formulas.
Finally, we shift our attention to functions with singularities. After proving some impor
tant theorems, we derive that our formulas converge for a specific class of singular functions.
The thesis concludes with some experiments, which compare the new formulas with the
corresponding Gauss formula and demonstrate the quality of our error bounds.
