Applications to Approximation Theory
There is a close connection between the classical approximation theory and the theory of interpolation spaces. We indicated this in 1.5. We discuss the link in more detail in the first two sections. In the first section, the main result is that every “approximation space” is a real interpolation space. The theorem makes the K-method (Chapter 3) available in approximation theory. This is then utilized in Section 2 to obtain, i. a., a classical theorem (of Jackson and Bernstein type; see 1.5) concerning the best approximation of functions in L p (ℝ n ) (1 ≤ p ≤ ∞) by entire functions of exponential type. In the following sections, 3 and 4, we prove other approximation theorems, using interpolation techniques developed in Chapter 3, 5 and 6. In particular, we treat approximation of operators by operators of finite rank, and approximation of differential operators by difference operators. Additional applications are indicated in Section 7.5 and 7.6, e.g., approximation by spline functions.
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