Lagrange and Hermite interpolation in banach spaces

AbstractLet f map a Banach space X into itself, and let x1, x2,…, xn be distinct points of X. Then there exists a polynomial y(x) of degree (n − 1) which interpolates f at these points. Furthermore, y(x) has a Lagrange representation y(x)=∑i=1n[w′i(xi)]−1wi(x)(x−xi)f(xi), where wi(x) = Li(x − x1, x − x2,…, x − xn), wi′(xi) is the first Fréchet derivative of wi at xi, and Li, i = 1, 2,…, n, is an appropriately chosen n-linear operator. In an analogous manner, an Hermite polynomial y(x) of degree (2n − 1) is derived, which interpolates f and f′ at x1, x2,…, xn. Finally, if X is a Hilbert space, the polynomials y(x) and y(x) are shown to have simple representations in terms of inner products