REGULARITY AND EXPLICIT REPRESENTATION OF (0,1,···, m-2, m) INTERPOLATION ON THE ZEROS OF (1-x~2)P_(n-2)~((α,β)) (x)

摘要

Necessary and sufficient conditions for the regularity and q-regularity of (0,1,··· ,m- 2,m) interpolation on the zeros of (1-x~2)P_(n-2)~((α,β))(x) (α,β > -1) in a manageable form are established, where P_(n-2)~((α,β))(x) stands for the (n-2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,··· ,m- 2,m) interpolation has an infinity of solutions then the general form of the solutions is f_0(x)+Cf(x) with an arbitrary constant C.