Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonai. We construct an orthogonal basis for the n-order（n ≥ 3） algebraic-trigonometric spline space in order to resolve the theo- retical problem that there is not an explicit orthogonai basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions, which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.
Let F be an algebraically closed field of prime characteristic p>3,and W(n)the Witt superalgebra over F,which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates.The dimensions of simple atypical modules in the restricted supermodule category for W(n)are precisely calculated in this paper,and thereby the dimensions of all simple modules can be precisely given.Moreover,the restricted supermodule category for W(n)is proved to have one block.
