We study the relationship of transformations between Legendre and Bernstein basis. Using the relationship, we present a simple and efficient method for optimal multiple degree reductions of Bézier curves with respect to the L2-norm.  2002 Elsevier Science B.V. All rights reserved.

We give an upper bound for the L condition number of the triangular Bernstein basis for polynomials of total degree at most n in s variables. The upper bound grows like (s + 1) when n tends to infinity. Moreover the upper bound is independent of s for s ≥ n− 1.

We prove that the exponential localization of a frame with respect to an orthonormal basis in a Hilbert space is not sufficient to get a Bernstein inequality. In other words, the fact that a function belongs to an approximation space of the frame cannot be characterized in terms of the sparseness of its frame coefficients.

Wavelet decomposition and its related nonlinear approximation problem are investigated on the basis of shift invariant spaces of functions. In particular, a Bernstein type inequality associated with wavelet decomposition is established in such a general setting. Several examples of piecewise polynomial spaces are given to illustrate the general theory. AMS Subject Classifications: 41 A 17, 41 A...

In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x, x] as linear combinations of Bernstein polynomials (x − x) · (x − x)n−k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the ...

We investigate the iterates of positive linear operators on C[0, 1] which preserve the linear functions, and the asymptotic behaviour of the associated semigroup. The general results are applied to the Bernstein-Schnabl operators on the unit interval.

Multiplier ideals are very important in higher dimensional geometry to study the singularities of ideal sheaves. It reflects the singularities of the ideal sheaves and provides strong vanishing theorem called the Kawamata-Viehweg-Nadel vanishing theorem (see [3]). However, the multiplier ideals are defined via a log resolution of the ideal sheaf and divisors on the resolved space, and it is dif...

We Produce a model of ZF+DC in which there are Bernstein sets, Luzin sets, and Sierpinski sets, but there is no Vitali sets and hence no Hamel basis. Definition. • B ⊂ ωω is called Bernstein iff B ∩ P 6= ∅ 6 = P \ B for all perfect P ⊂ ωω. • Letting E0 ⊂ (ωω)2 be the Vitali equivalence relation defined by xE0y iff ∃n0∀n ≥ n0, x(n) = y(n), V ⊂ ωω is called Vitali if V picks exactly one element f...

A new differential-recurrence relation for the B-spline functions of same degree is proved. From this relation, a recursive method computing coefficients m in Bernstein–Bézier form derived. Its complexity proportional to number case coincident boundary knots. This means that, asymptotically, algorithm optimal. In other cases, increased by at most O(m3). When basis are known, it possible compute...