Intro to Analysis in-class_Part_58

Intro to Analysis in-class_Part_58 - 7.5 Approximation by...

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7.5 Approximation by polynomials 121 Convolution is a smoothing operation. Theorem 7.5.5. For f, g ∈ R ( D ) , f * g is continuous, D compact. Proof. First, prove it for the case when f, g are continuous. Fix ε > 0 and c D . Compact support gives a bound for the continuous function f , say | f ( x ) | < B . Also, it gives uniform continuity of g , so find δ > 0 such that | x - c | < δ = ⇒ | ( x - y ) - ( c - y ) | < δ = ⇒ | g ( x - y ) - g ( c - y ) | < ε. Now for | x - c | < δ , we have | ( f * g )( x ) - ( f * g )( c ) | = fl fl fl fl Z D f ( y ) g ( x - y ) - g ( c - y ) dy fl fl fl fl Z D | f ( y ) | · | g ( x - y ) - g ( c - y ) | dy · | spt f | . Now suppose that f, g are not necessarily continuous. Let { f n } ⊆ C ( D ) be a sequence with | f n ( x ) | ≤ B and R D | f n ( x ) - f ( x ) | dx n →∞ -----→ 0, and similarly { g n } ⊆ C ( D ). Then f * g - f n * g n = ( f - f n ) * g + f n * ( g - g n ) . Consequently, we have | (( f - f n ) * g )( x ) | ≤ Z D | f ( x - y ) - f n ( x - y ) | · | g ( y ) | dy sup y D | g ( y ) | Z D | f ( x - y ) - f n ( x - y ) | dy n →∞ -----→ 0 . Note that this actually shows sup x D | (( f - f n ) * g )( x ) | → 0, so that ( f - f n ) * g unif ----→ 0 on D . Similarly, f n * ( g - g n ) unif ----→ 0 on D . Since each f n * g n is continuous, the uniform limit f * g is, too.
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