Gelfond-Bezier曲线
Gelfond-Bezier Curves
AI总结：
本文通过将Müntz空间中的广义Bernstein基表示为Chebyshev-Bernstein基在区间端点趋于零时的极限，将曲线设计中的de Casteljau算法、开花和维数提升等概念推广到Gelfond-Bernstein基，从而简化了相关算法。
摘要：
我们证明，由Hirschman和Widder [7]定义并由Gelfond [6]推广的Müntz空间中的广义Bernstein基，可以表示为区间[a,1]上Müntz空间中Chebyshev-Bernstein基当实数a趋近于零时的极限。这种实现使得曲线设计中的概念，如de Casteljau算法、开花、维数提升，能够从Müntz空间中Chebyshev开花的一般理论转移到这些广义Bernstein基上，我们将其称为Gelfond-Bernstein基。与Chebyshev-Bernstein基相比，使用Gelfond-Bernstein基的优势在于所获得的概念和算法更为简单。
英文摘要：
We show that the generalized Bernstein bases in Muntz spaces defined by Hirschman and Widder [7] and extended by Gelfond [6] can be obtained as limits of the Chebyshev-Bernstein bases in Muntz spaces with respect to an interval [a,1] as the real number, a, converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be translated from the general theory of Chebyshev blossom in Muntz spaces to these generalized Bernstein bases that we termed here as Gelfond-Bernstein bases. The advantage of working with Gelfond-Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev-Bernstein bases counterparts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
