On the approximation by {polynomials

As usual, p∗ is called a best approximation (b.a.) to f in (or, by elements of) IPγ,n. To give some examples, let X = Lp[0, 1] and set γ(t) = G(·, t), where G(s, t) is defined on [0, 1] × T . With G Green’s function for a k–th order ordinary linear initial value problem on (0, 1] and T = [0, 1), one has approximation by generalized splines. With G(s, t) = e and T = IR, one has approximation by exponential sums. With G(s, t) = (1 + st)−1 and T = (−1,∞), one has an approximation problem which shares many features with approximation by rational functions. The last two examples lend themselves easily to an extension of T to the complex plane, and reveal their essential properties only after such an extension has been made [5]. Other examples may be found in [7]. A seemingly different example occurs in Numerical Analysis. Here X is the topological dual Y ∗ of a normed linear space Y of functions defined on T , and, for t ∈ T , γ(t) is the linear functional of point evaluation at t, i.e., for all y ∈ Y, γ(t)y = y(t). Best approximation by γ–polynomials of order n to f ∈ X amounts to the construction of a best approximate rule of the form ∑n i=1 aiy(ti) for the evaluation of the linear functional f at y. But it is not difficult to see that many approximation problems by γ–polynomials of fixed order can be considered to be special cases of the last example. For, with X, γ and T given, let Y be the linear space of functions on T whose general element y is given by