Some Questions of Hard Analysis
In the theory of functions on ℝN there is a great chasm between the space of C∞ functions and the space of real analytic functions. If, for instance, a real analytic function vanishes on a set of positive measure, then it is identically zero. [This is most easily proved by induction on dimension, beginning with the fact that in dimension 1 we have the stronger result that if the zero set has an interior accumulation point, then the function is identically zero.] By contrastany closed set is the zero set of a C∞ function. In dimension 1 this is seen by noting that the complement of the closed set is the disjoint union of open intervals; it is straightforward to construct a C∞ function of compact support on the closure of an open interval whose support is precisely that closed interval. In several real variables the Whitney decomposition serves as a substitute for the interval decomposition of an open set and, with more effort, allows a similar construction to be effected (see Proposition 3.3.6).
KeywordsReal Root Real Variable Real Analytic Function Infinite Product Taylor Coefficient
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