Polynomial Interpolation - Convergence Properties

Convergence Properties

It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converges to the interpolated function as the degree n goes to infinity? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm.

The situation is rather bad for equidistant nodes, in that uniform convergence is not even guaranteed for infinitely differentiable functions. One classical example, due to Carl Runge, is the function f(x) = 1 / (1 + x2) on the interval . The interpolation error ||f − p_n|| grows without bound as . Another example is the function f(x) = |x| on the interval, for which the interpolating polynomials do not even converge pointwise except at the three points x = −1, 0, and 1.

One might think that better convergence properties may be obtained by choosing different interpolation nodes. The following theorem seems to be a rather encouraging answer:

For any function f(x) continuous on an interval there exists a table of nodes for which the sequence of interpolating polynomials converges to f(x) uniformly on .

Proof. It's clear that the sequence of polynomials of best approximation converges to f(x) uniformly (due to Weierstrass approximation theorem). Now we have only to show that each may be obtained by means of interpolation on certain nodes. But this is true due to a special property of polynomials of best approximation known from the Chebyshev alternation theorem. Specifically, we know that such polynomials should intersect f(x) at least n+1 times. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial.

The defect of this method, however, is that interpolation nodes should be calculated anew for each new function f(x), but the algorithm is hard to be implemented numerically. Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function f(x)? The answer is unfortunately negative as it is stated by the following theorem:

For any table of nodes there is a continuous function f(x) on an interval for which the sequence of interpolating polynomials diverges on .

The proof essentially uses the lower bound estimation of the Lebesgue constant, which we defined above to be the operator norm of X_n (where X_n is the projection operator on Π_n). Now we seek a table of nodes for which

Due to the Banach–Steinhaus theorem, this is only possible when norms of X_n are uniformly bounded, which cannot be true since we know that

For example, if equidistant points are chosen as interpolation nodes, the function from Runge's phenomenon demonstrates divergence of such interpolation. Note that this function is not only continuous but even infinitely times differentiable on . For better Chebyshev nodes, however, such an example is much harder to find because of the theorem:

For every absolutely continuous function on the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly.