Intro to Analysis in-class_Part_59

Intro to Analysis in-class_Part_59 - 7.5 Approximation by...

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7.5 Approximation by polynomials 123 (iii) For every δ > 0, lim n →∞ R | x |≥ δ g n ( x ) = 0. If f * g n is the weighted average of f ( y - x ), with weights given by g n , then (iii) shows that g n concentrates all weight at the origin as n → ∞ . In fact, (iii) is equivalent to (iii’) δ > 0 ,n >> 1 = spt g n ( - δ,δ ) . It turns out that the Dirac delta δ is the identity with respect to convolution: f * δ = δ * f = f , for any f . An approximate identity is the best one can do at a continuous approximation of δ . Even though f * g n 6 = f , we do have the following. Theorem 7.5.10. Let { g n } be an approximate identity, and let f ∈ R ( I ) . Then f is continuous at x = lim n →∞ ( f * g n )( x ) = f ( x ) . If f is continuous, then the limit is uniform. Proof. If ε > 0 and f is continuous at x , choose δ > 0 such that | y | < δ = ⇒ | f ( x - y ) - f ( x ) | < ε. Then by the properties of the approx identity,
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_59 - 7.5 Approximation by...

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