Abstract. We prove that if f is increasing on [ -1,1], then for each n = 1, 2 ..... there is an increasing algebraic polynomial P. of degree n such that {f(x) -P.(x){ < cw2( f, V/I -x 2/n), where w2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.

Abstract.We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

Based on previous work for the static problem, in this paper, we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.

ABSTRACT. We obtain various estimates for the error in adaptive approximation and also establish a relationship between adaptive approximation and free-knot spline approximation.

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