Almost Strictly Total Positivity of AH B-Spline Bases

Almost strictly total positivity is highly related with the poisedness of curve interpolation problem.It is living a basic status in the bases theory.A geometrical approach is proposed to prove the collection matrices of AH B-spline bases are almost strictly totally positive matrices.Firstly,AH B-spline function is introduced.Using knot inserted algorithm,the relationship between the number of zero points and the number of coefficient variations can be deduced.Finally,using mathematical induction,almost strictly total positivity of AH B-spline bases can be proved.The approach is intuitive and geometrical.