By A.M. Fink

ISBN-10: 3540067299

ISBN-13: 9783540067290

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**Example text**

Proof. Let E > 0. Find an open neighbourhood U of {(x, x) : x E X} such that (x,t) EU implies that lhz(t)I = lh(x,t)I < E. Then (Xx X) \ U is a compact min P:i:(t). Since (X x X) \ U is compact, m > 0. Let set. Let m := te(XxX)\U M = max{lh(x, t)I : (x, t) E X x X}. Now if (x, t) E U then lhz(t)I < E = ee(t). Otherwise, M lhz(t)I $ M $ m P:i:(t). Thus, for every (x, t) E X x X, we have lhz(t)I < E e(t) - M + -Pz(t). m So, M ILn(hz,x)I < ELn(e,x) + -Ln(Pz,x). - m Hence, using Lemma 1, we obtain Since E > 0 is arbitrary, this proves the Lemma.

Indeed, since A separates points of X, there is a function g EA such that g(x) "I- g(y). Also since A vanishes at no point of X, there are functions h and k in A such that h(x) -=f. 0 and k(y) -=f. 0. Let u = gk - g(x)k and v = gh - g(y)h. Then u E A,v E A,u(x) = v(y) = O,u(y) -=f. 0, and v(x) -=f. 0. Let f = av v(x) Clearly f + satisfies the required properties. (Ju . u(y) • We now consider the complex case. The following simple example shows that Theorem 3 is not valid with C(X, JR) replaced by C(X, «::).

G. [T-L, Ru3]). Carleson's theorem is proved in [Ca] and its extension in [Hu]. Fejer's theorem originated in [Fel] and this was amplified further in [Fe2]. The notion of approximate identity is a well known concept in harmonic analysis (cf. , [Ho]). The Weierstrass theorem was proved independently by Weierstrass [We] and Runge [Runl, Run2]. A number of other proofs of the theorem appeared subsequently (cf. [Pi, Vo, Lel, Mit, Fel, Ler, L, Del]). Bernstein's theorem was proved in [Bernl]. For a detailed study of many interesting properties of Bernstein polynomials, see [Lorl].

### Almost Periodic Differential Equations by A.M. Fink

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