Negative Theorems in Approximation Theory

1. INTRODUCTION. Negative theorems have a rich tradition in mathematics. In fact mathematics seems to be unique among the sciences in that negative results are very much a part of the mathematical edifice. Most mathematical theories try to explain what is possible and also what is not. To understand the structure of a mathematical theory is also to understand its limitations. There are many different types of negative theorems. Some simply say that something is impossible. For example, one of the classical problems of Greek mathematics asks whether it is possible, using only a ruler and a compass, to square the circle, i.e., to construct a square with the same area as a given circle. This was answered in the negative in 1882 when Lindemann proved that π is transcendental. (Lambert had already proved that π is irrational, but this is insufficient to prove the impossibility of squaring the circle.) We recall that Abel established the insolvability of the general quintic equation, i.e., one cannot find a formula for the roots of fifth degree polynomi-als that involves only the coefficients of the polynomials and radicals. And, of course, there is Wiles's proof of Fermat's Last Theorem. Each of these problems and results is a fundamental part of our mathematical heritage. Each has had a profound influence. Nevertheless, the results themselves simply say that something is impossible. Other negative results delineate what cannot be done, as compared with what is possible. The best of these results also highlight the salient features that illustrate why they are not possible. For example, Gauss proved that a necessary and sufficient condition for the construction of a regular n-gon, using only ruler and compass, is that

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