A new geometric method to prove the total positivity of UE spline basis was proposed. UE spline basis is a kind of basis defined over algebraic-trigonometric unified space. UE spline basis shares most properties of th...A new geometric method to prove the total positivity of UE spline basis was proposed. UE spline basis is a kind of basis defined over algebraic-trigonometric unified space. UE spline basis shares most properties of the usual polynomial B-Splines. Total positivity is an important property for spline basis, it is highly related with shape preserving and variation diminishing properties. Knot inserted algorithm is the most useful algorithm for spline curves since many other useful properties are based on it. It is necessary to prove the total positivity of UE spline basis using knot inserted algorithm intuitively, not only enrich the UE spline basis theory, but also can be treated as supplement to the total positivity in algebraic sense. This approach also can be extended to other analogical bases.展开更多
The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves...The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.展开更多
基金Supported by the National Science Foundation of China (60970079 and 60933008)
文摘A new geometric method to prove the total positivity of UE spline basis was proposed. UE spline basis is a kind of basis defined over algebraic-trigonometric unified space. UE spline basis shares most properties of the usual polynomial B-Splines. Total positivity is an important property for spline basis, it is highly related with shape preserving and variation diminishing properties. Knot inserted algorithm is the most useful algorithm for spline curves since many other useful properties are based on it. It is necessary to prove the total positivity of UE spline basis using knot inserted algorithm intuitively, not only enrich the UE spline basis theory, but also can be treated as supplement to the total positivity in algebraic sense. This approach also can be extended to other analogical bases.
文摘The present paper finds out that the geometric entity which characterizes the best Lipschitz constants for the Bezier nets and Bernstein polynomials over a simplex sigma is an angle Phi determined by sigma, and proves that (1) if f(x) is Lipschitz continuous over sigma, i.e., f(x) is an element of Lip(A)(alpha,sigma), then both the n-th Bezier net <(f)over cap (n)> and the n-th Bernstein polynomial B-n(f;x) corresponding to f(x) belong to Lip(B)(alpha,sigma) , where B = Asec(alpha)Phi; and (2) if n-th Bezier net <(f)over cap (n)> is an element of Lip(A)(alpha,sigma), then the elevation Bezier net <E(f)over cap (n)> and the corresponding Bernstein polynomial. B-n(f,;x) also belong to Lip(A)(alpha,sigma). Furthermore, the constant B = Asec(alpha)Phi, in case (1) is best in some sense.
