Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation for all 1 <= p, q <= infinity,0 <= s <= r -1, parallel to(f - L(r)f)((s))parallel to(p) <= C(r, s, p, q) (b - a)(r-s+1/p-1/q)parallel to f((r))parallel to q, where f is an element of W-q(r)[a,b] and L-r is the Birkhoff interpolation based on r pairs of numbers (x(i), k(i))(i=1)(r), with its Polya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of C(r, s, p, q) to the norm of an integral operator. Second, we refer the values of C(r , s , 1, 1) and C(r, s, infinity, infinity) to two explicit integral expressions and the value of C(r, s,2,2) to the computation of the maximum eigenvalue of a Hilbert-Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality (s = 0) and sharp Picone inequality (1 <= s <= r - 1).