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NEW DOCTORAL DEGREE Montgomery identity, quadrature formulae and derived inequalities
Mihaela Ribičić- Penava
; Department of Mathematics, University of Osijek, Osijek, Croatia
Abstract
The aim of this PhD dissertation is to
give generalizations of classical quadrature formulae with two,
three and four nodes using some generalizations of the weighted
Montgomery identity. Thereby families of weighted and non-weighted
quadrature formulae are considered, some error estimates are
derived, and sharp and the best possible inequalities as well as
Ostrowski type inequalities are proved.
Classes of weighted and non-weighted two-point
quadrature formulae are studied and corresponding error estimates
are calculated. Two-point Gauss-Chebyshev formulae of the first
and of the second kind as well as genera\-lizations of the
trapezoidal formula, Newton-Cotes two-point formula, Maclaurin
two-point formula and midpoint formula are obtained as special
cases of these formulae.
The dissertation deals with three-point quadrature formulae,
generalizations of Simpson's, dual Simpson's and Maclaurin's
formula, three-point Gauss-Chebyshev formulae of the first kind
and of the second kind that follow from a general formula, as well
as corresponding error estimates.
It is also dedicated to closed four-point quadrature formulae
from which a we\-ight\-ed and non-weighted generalization of Bullen
type inequalities for $(2n)-$ convex functions is obtained. As
a special case, Simpson's $3/8$ formula and Lobatto four-point
formula with related inequalities are considered.
Weighted Euler type identities, which
represent weighted integral one-point formulae, are worked out in the dissertation as well. By means of these
identities, generalized weighted quadrature formulae are derived
in which the integral is estimated by function values in $n$ nodes
and generalizations of Gauss-Chebyshev formulae of the first and
of the second kind are given. Error estimates are derived and some
sharp and best possible inequalities are proved for all given
formulae.
Keywords
Hrčak ID:
53709
URI
Publication date:
3.6.2009.
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