Central Subspaces of Banach Spaces

In this paper we study Banach spaces that admit weighted Chebyshev centres for finite sets. Such spaces have been extensively studied recently by Veselý using the approach of finitely intersecting balls. Following his approach we exhibit large classes of Banach spaces that have this property. Certain stability results for spaces of vector valued continuous and Bochner integrable functions are also obtained.     

SINGULAR INTEGRAL OPERATORS IN L2 SPACE WITH CHEBYSHEV WEIGHTS

1IntroductionLetL:=L:l-1,11bethefunctionspacewhichconsist8offunctionsfsatisfyingwherep(t)isaweightfunction,denotethenormthenwecanknowfroml11that.hereexistsanorthogonalsetofpolynodrialsyto,yt1,...,ytnt...andEspeciallyweknowfrom[1]thatifweightfunctionp(f)maybechosenasw1(t)=(1-t')-t,wz(t)=(1-f')i,ws(t)=(1-t)t(1 t)-1,w`(t)=(1-f)-t(1 t)i,thecorrespondingrespectiveorthogonalpolynondalsareTJ')(t)=cosno,Tj')(t)=sin(n 1)o/sino,T1')(t)=sinngo/sint,T;')(f)=cosnge/cos9,wheret=coso,thenby(1.3)theyarethed…     

Blossoming: A Geometrical Approach

A geometrical approach of a notion of blossom for piecewise smooth Chebyshev functions is developed by considering convenient intersections of osculating flats. A subblossoming principle allows us to obtain all the expected properties and leads to the notion of blossom for splines based on a given piecewise smooth Chebyshev function. January 7, 1997. Date revised: October 1, 1997. Date accepted: December 22, 1997.     

An Accurate Solution of the Poisson Equation by the Finite Difference-Chebyshev-Tau Method

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Sets with External Chebyshev Layer

Mathematical Notes -     

Alexandroff's Double Arrow Compact Space and Approximation Theory

A compact space Q similar to the compact space known as Alexandroff's double arrow space is constructed. It is shown that the real space C(Q) has no Chebyshev subspaces of codimension >1, but the complex space C(Q) has such subspaces.     

A Modification of Lagrange Interpolation

A modification of Lagrange interpolation based on the zeros of the Chebyshev polynomial of the second kind is constructed, which interpolates at many ofgiven data. Thus, for this node-system the main result gives an affimative answer to a problem suggested by Bernstein in 1930. Moreover, our modification has a Timan-Gopengauz type approximation rate.     

Error Analysis of Clenshaw's Algorithm for Evaluating Derivatives of a Polynomial

A forward rounding error analysis is presented for the extended Clenshaw algorithm due to Skrzipek for evaluating the derivatives of a polynomial expanded in terms of orthogonal polynomials. Reformulating in matrix notation the three-term recurrence relation satisfied by orthogonal polynomials facilitates the estimate of the rounding error for the m-th derivative, which is recursively estimated in terms of the one for the (m – 1)-th derivative. The rounding errors in an important case of Chebyshev polynomial are discussed in some detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer--van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.