Dyn, N., E. Farkhi, A. Mokhov.
Approximation of Univariate Set-Valued
Functions - an Overview
(pp. 495-514)
A B S T R A C T S
COMPLEX ANALOGUES OF THE ROLLE'S THEOREM
Bl. Sendov
acad@sendov.com
2000 Mathematics Subject Classification: 30C10.
Key words:
Complex Rolle's theorem.
Classical Rolle's theorem and its analogues for complex algebraic polynomials
are discussed. A complex Rolle's theorem is conjectured.
SMALE'S CONJECTURE ON MEAN VALUES OF POLYNOMIALS AND
ELECTROSTATICS
Dimitar K. Dimitrov
dimitrov@ibilce.unesp.br
2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.
Key words:
Zeros of polynomials, critical
points, Smale's conjecture, extremal problem, electrostatics.
A challenging conjecture of Stephen Smale on geometry of polynomials
is under discussion. We consider an interpretation which turns out
to be an interesting problem on equilibrium of an electrostatic
field that obeys the law of the logarithmic potential. This
interplay allows us to study the quantities that appear in Smale's
conjecture for polynomials whose zeros belong to certain specific
regions. A conjecture concerning the electrostatic equilibrium
related to polynomials with zeros in a ring domain is formulated and
discussed.
SOBOLEV TYPE DECOMPOSITION OF PALEY-WIENER-SCHWARTZ SPACE
WITH APPLICATION TO SAMPLING THEORY
Dimiter
Dryanov
ddryanov@alcor.concordia.ca
2000 Mathematics Subject Classification: 94A12, 94A20, 30D20, 41A05.
Key words:
Paley-Wiener-Schwartz space, Shannon
sampling theorem, Tschakaloff-Bernstein representation formulas,
Levin transcendental interpolating theory.
We characterize Paley-Wiener-Schwartz space of entire functions as a
union of three-parametric linear normed subspaces determined by the
type of the entire functions, their polynomial asymptotic on the
real line, and the index p ³ 1 of a Sobolev type
L_{p}-summability on the real line with an appropriate weight
function. An entire function belonging to a sub-space of the
decomposition is exactly recovered by a sampling series, locally
uniformly convergent on the complex plane. The sampling formulas
obtained extend the Shannon sampling theorem, certain representation
formulas due to Bernstein, and a transcendental interpolating theory
due to Levin.
METHODS WITH A SPARSE JACOBIAN FOR
SOLVING NONLINEAR SYSTEMS OF EQUATIONS
Nikolay Kyurkchiev
nkyurk@math.bas.bg,
Anton Iliev
aii@uni-plovdiv.bg
2000 Mathematics Subject Classification:
65H10.
Key words:
Nonlinear systems of equations, numerical solution,
Iliev's, Halley's and Euler-Chebyshev's methods,
fixed-point relations.
Here we give methodological survey of contemporary methods for
solving nonlinear systems of equations in R^{n}. The reason of this
review is that many authors in present days rediscovered such
classical methods. In particular, we consider Newton's-type
algorithms with sparse Jacobian. Method for which the inverse matrix
of the Jacobian is replaced by the inverse matrix of the
Vandermondian is proposed. A number of illustrative numerical
examples are displayed. We demonstrate Herzberger's model with
fixed-point relations to the some discrete versions of Halley's and
Euler-Chebyshev's methods for solving such kind of systems.
SOME COEFFICIENT ESTIMATES FOR POLYNOMIALS ON THE UNIT
INTERVAL
M. A. Qazi
qazima@aol.com,
Q. I. Rahman
rahmanqi@dms.umontreal.ca
2000 Mathematics Subject Classification:
26C05, 26C10, 30A12, 30D15, 42A05, 42C05.
Key words:
Polynomials, Inequality, Weighted L^{p} norm.
In this paper we present some inequalities about the moduli of the
coefficients of polynomials of the form
f (x) : = å_{n = 0}^{n}a_{n} x^{n}, where a_{0}, ¼, a_{n} Î C. They can be
seen as generalizations, refinements or analogues of the famous
inequality of P. L. Chebyshev, according to which
|a_{n}| £ 2^{n-1} if
| å_{n = 0}^{n} a_{n} x^{n} | £ 1 for -1 £ x £ 1.
EQUIVALENCE BETWEEN K-FUNCTIONALS BASED ON CONTINUOUS
LINEAR TRANSFORMS
Borislav R. Draganov
bdraganov@fmi.uni-sofia.bg,
Kamen G. Ivanov
kamen@math.bas.bg
2000 Mathematics Subject Classification:
46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.
Key words:
K-functional, modulus of smoothness, rate of convergence, best
approximation, linear operator.
The paper presents a method of relating two K-functionals by means
of a continuous linear transform of the function. In particular, a
characterization of various weighted K-functionals by unweighted
fixed-step moduli of smoothness is derived. This is applied in
estimating the rate of convergence of several approximation
processes.
APPROXIMATION OF UNIVARIATE SET-VALUED FUNCTIONS - AN
OVERVIEW
Nira Dyn
niradyn@soul.cs.tau.ac.il,
Elza Farkhi
elza@post.tau.ac.il,
Alona Mokhov
alonamok@post.tau.ac.il
2000 Mathematics Subject Classification:
26E25, 41A35, 41A36, 47H04, 54C65.
Key words:
compact sets, set-valued
functions, linear approximation operators, Minkowski sum of sets,
metric average, metric linear combinations.
The paper is an updated survey of our work on the approximation of
univariate set-valued functions by samples-based linear
approximation operators, beyond the results reported
in our previous overview.
Our approach is to adapt operators for
real-valued functions to set-valued functions, by replacing
operations between numbers by operations between sets. For
set-valued functions with compact convex images we use Minkowski
convex combinations of sets, while for those with general compact
images metric averages and metric linear combinations of sets are
used. We obtain general approximation results and apply them to
Bernstein polynomial operators, Schoenberg spline operators and
polynomial interpolation operators.
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