By William F. Donoghue
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Additional resources for Distributions and Fourier transforms
26 I. INTRODUCTION (3) K , c K , implies m(K,)5 m(K,). (4) If K c , K , , then m(K)S m(K,). (5) If K, n K, = 0 , then m(K,u K,) = m(K,)+ m(K,). ur= , To establish (9,one should note that there exist positive functions in C , ( X ) vanishing on the one compact set and equal to + 1 on the other. We next define a set function p(G) on the open subsets of X by setting p(G) = sup m(K), K c G . It is not hard to check that p has generally the same properties: + 0 5 p(G) 5 rn for all G. (2) P ( 0 ) = 0.
Theorem: Let f(x) be C" on (c, d ) such that for every point x in the interval there exists an integer N , for which f'"-'(x) = 0; then f ( x ) is a polynomial. PROOF: Let G be the open set of all x for which there exists a neighborhood within which f ( x ) coincides with some polynomial, that is, a neighborhood on whichf'k'(x) vanishes identically for some value of k. Let F be the complement of G ; the theorem will be proved if we show that F is empty. The set F cannot have an isolated point, for if x, were such a point, it is the right-hand endpoint of an interval (a, x,) on which f coincides with a polynomial, and the Taylor coefficients of the polynomial coincide with the formal Taylor expansion off(x) about x,; similarly, x, is the left-hand endpoint of an interval (x,, b) on whichf(x) coincides with a polynomial, which is determined by the Taylor expansion of the function about x,; thus,fcoincides with a certain polynomial in the interval (a, b) and x, is not in F.
Theorem: Let X be a locally compact metric space and C , ( X ) the linear space of all continuous functions on X which vanish outside a compact subset of that space. If F(f)is a linear functional on C,(X) having the property that F(f)2 0 wheneverf(x) 2 0, then there exists a Radon measure p on X such that for allf PROOF: We first construct the measure p , defining the function m(K) on the class of compact subsets of X as follows: m ( K ) = inf F(u), u(x) 2 0 on X , u(x) 2 1 on K . It is fairly easy to verify that m(K) has the following properties: ( I ) 0 5 m(K) < co for all K .
Distributions and Fourier transforms by William F. Donoghue