By J. Prolla

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Additional info for Approximation Theory and Functional Analysis

Example text

On the other hand, U o is possibly not a nearest point to other elements on the ray {It: t > 1}, see Fig. 4. 1 Definition. An existence set M in a normed linear space is a sun if, given ! ¢ M, there is a Uo E PM! such that Uo is also a nearest point to ft = U o + t(f - uo) Fig. 3 for each t > 1. 32 II. Nonlinear Approximation: The Functional Analytic Approach Fig. 4. A set which is not a sun A nearest point with this property is said to be a solar point to f If each U o E PMI is a solar point to I, then M is said to be a strict sun.

D In a uniformly convex space E, we get extra information on suns. Let f E E\M and U o be a solar point to f. Let r = d(f, M). Given J > 0, we know that Pol E Br+fJ(f)\B 2r (f2). 5, diam(Pd) tends to zero as J -+ o. Fig. 6. 2 34 II. Nonlinear Approximation: The Functional Analytic Approach Hence f E TiJ and TiJ = E. Therefore, each sun in a uniformly convex set is an approximatively compact Chebyshev set. B. 5 Theorem. The following conditions on a normed linear space E are equivalent: 1°. E is smooth.

Critical Points (2) Let M = {(X,Y)E ~2: x ~ 0, lyl ~ x 2 }. Then CoM = {(X,0)E~2: x ~ O}. This elucidates that a tangent cone of a two-dimensional space may collapse to a one dimensional set, though CoM = KoM.