164 research outputs found
Interpolation by Linear Functions on an -Dimensional Ball
By we denote the Euclidean ball in given by
the inequality . Here ,
. We mean by the space of
continuous functions with the norm
and by
the set of polynomials in variables of degree , i.e., linear
functions on . Let be the vertices
of -dimensional nondegenerate simplex . The interpolation
projector
corresponding to is defined by the
equalities We obtain the
formula to compute the norm of as an operator from into via
, and coefficients of basic Lagrange polynomials of . In more
details we study the case when is a regular simplex inscribed into
.Comment: 17 pages, 6 figures, 1 tabl
Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball
Let be the Euclidean unit ball in given by the
inequality , . By we mean the space of continuous
functions with the norm . The symbol
denotes the set of polynomials in variables of degree , i.e., the
set of linear functions upon . Assume are the vertices of an -dimensional nondegenerate simplex
. The interpolation projector
corresponding to is defined by the equalities Denote by the norm of as an operator
from onto . We describe the approach in which
can be estimated from below via the volume of .Comment: 10 page
On Properties of a Regular Simplex Inscribed into a Ball
Let be a Euclidean ball in and let be a space
of~continuous functions with the uniform norm
By we
mean a set of polynomials of degree , i.e., a set of linear functions
upon . The interpolation projector with the nodes is defined by the equalities
, . The
norm of as an operator from to can be calculated by the
formula Here are the
basic Lagrange polynomials corresponding to the -dimensional nondegenerate
simplex with the vertices . Let be a projector having
the nodes in the vertices \linebreak of a regular simplex inscribed into the
ball. We describe the points with the property . Also we formulate a geometric conjecture which implies that
is equal to the minimal norm of an interpolation projector
with nodes in . We prove that this conjecture holds true at least for
.
Keywords: regular simplex, ball, linear interpolation, projector, normComment: 13 page
The Minimum Norm of a Projector under Linear Interpolation on a Euclidean Ball
We prove the following proposition. Under linear interpolation on a Euclidean
-dimensional ball , an interpolation projector whose nodes coincide with
the vertices of a regular simplex inscribed into the boundary sphere has the
minimum -norm. This minimum norm is equal to
, where
, , and
. For any
, Moreover,
only for and if and only if
is an integer.Comment: 7 page
On a Geometric Approach to the Estimation of Interpolation Projectors
Suppose is a closed bounded subset of is an
-dimensional non-degenerate simplex, . Here is the result of
homothety of with respect to the center of gravity with coefficient
. Let be linearly
independent monomials in variables,
Put The interpolation projector with a set of nodes is defined by
equalities Denote by
the norm of as an operator from to
. Consider the mapping of the
form We have the following
inequalities: Here is
the -dimensional simplex with vertices We
discuss this and other relations for polynomial interpolation of functions
continuous on a segment. The results of numerical analysis are presented.Comment: 13 pages, 2 figure
On Some Estimate for the Norm of an Interpolation Projector
Let be the unit cube in and let be a space of continuous functions with the norm By denote a set of polynomials in variables of degree , i. e., a set of linear functions on . The interpolation projector with the nodes is defined by the equalities , . Let be the norm of as an operator from to . If is an Hadamard number, then there exists a non-degenerate regular simplex having the vertices at vertices of . We discuss some approaches to get inequalities of the form for the norm of the corresponding projector
Microstructure and properties of hypoeutectic silumin treated by high-current pulsed electron beams
The structural-phase states, microhardness, and tribological properties of hypoeutectic silumin after electron-beam treatment are studied by the methods of contemporary physical materials science. The object of the study is hypoeutectic АК10М2Н-type silumin containing 87.88 wt.% of Al and 11.1 wt.% of Si as the base components.Методами сучасного фізичного матеріалознавства досліджено структурно-фазові стани, мікротвердість і трибологічні властивості доевтектичного силуміну після електронно-пучкового оброблення. Об єктом дослідження був доевтектичний силумін марки АК10М2Н із вмістом 87,88 ваг.% Al й 11,1 ваг.% Si як головних компонентів.Методами современного физического материаловедения исследованы структурнофазовые состояния, микротв рдость и трибологические свойства доэвтектического силумина после электронно-пучковой обработки. Объектом исследования являлся доэвтектический силумин марки АК10М2Н с содержанием 87,88 вес.% Al и 11,1 вес.% Si как главных компонентов
О гипотезе Лассака для выпуклого тела
In 1993 M. Lassak formulated (in the equivalent form) the following conjecture. If we can inscribe a translate of the cube into a convex body , then . Here denotes the width of in the direction of the ith coordinate axis. The paper contains a new proof of this statement for n = 2. Also we show that if a translate of can be inscribed into the n-dimensional simplex, then for this simplex holds .В 1993 г. М. Лассак сформулировал (в эквивалентном виде) следующую гипотезу. Если в выпуклое тело можно вписать транслят куба , то . Здесь - ширина в направлении i-й координатной оси. В статье даётся новое доказательство этого утверждения для n = 2. Также мы показываем, что для n-мерного симплекса, в который можно вписать транслят , справедливо
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