Weighted Best Local Approximation

The mobility of a linkage is determined by the constraints imposed on its members. The constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The instantaneous motions are determined by the constraints, rather than by the c-space geometry. Shaky linkages are prominent examples that exhibit a higher instantaneous than finite DOF even in regular configurations. Inextricably connected to the mobility are kinematic singularities that are reflected in a change of the instantaneous DOF. The local analysis of a linkage, aiming at determining the instantaneous and finite mobility in a given configuration, hence needs to consider the c-space geometry as well as the constraint system. A method for the local analysis is presented based on a higher-order local approximation of the c-space adopting the concept of the tangent cone to a variety. The latter is the best local approximation of the c-space in a general configuration. It thus allows for investigating the mobility in regular as well as singular configurations. Therewith the c-space is locally represented as an algebraic variety whose degree is the necessary approximation order. In regular configurations the tangent cone is the tangent space. The method is generally applicable and computationally simple. It allows for a classification of linkages as overconstrained and underconstrained, and to identify singularities.

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Best Local Approximation in Orlicz Spaces

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2011.590264 ◽

2011 ◽

Vol 32 (11) ◽

pp. 1127-1145 ◽

Cited By ~ 3

Author(s):

H. Cuenya ◽

F. Levis ◽

M. Marano ◽

C. Ridolfi

Keyword(s):

Orlicz Spaces ◽

Local Approximation ◽

Best Local Approximation

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ALTERNATION THEOREM FOR $C(I,X)$ AND APPLICATION TO BEST LOCAL APPROXIMATION

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.24.1993.4483 ◽

1993 ◽

Vol 24 (2) ◽

pp. 135-147

Author(s):

A. AL-ZAMEL ◽

R. KHALIL

Keyword(s):

Banach Space ◽

Approximation Property ◽

Local Approximation ◽

Continuous Functions ◽

Space Of Continuous Functions ◽

Best Local Approximation ◽

Alternation Theorem

Let $X$ be a Banach space with the approximation property, and $C(I,X)$ the space of continuous functions defined on $I = [0,1)$ with values in $X$. Let $u_i \in C(I,X)$, $i=1,2,\cdots, n$ and $M=span\{u_1, \cdots, u_n\}$. The object of this paper is to prove that if $\{u_1, \cdots, u_n\}$ satisfies certain conditions, then for $f \in C(I,X)$ and $g \in M$ we have $||f-g|| = \inf\{||f-h|| : h\in M\}$ if and only if $f-g$ has at least $n$-zeros. An application to best local approximation in $C(I,X)$ is given.

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