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 ) | = Z D f ( y ) g ( x - y ) - g ( c - y ) dy 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
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Intro to Analysis in-class_Part_58 - 7.5 Approximation by...

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