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Intro to Analysis in-class_Part_57

# Intro to Analysis in-class_Part_57 - the rest of this...

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7.4 Power series 119 Theorem 7.4.10. If f is analytic and f ( x ) = 0 , x ( a,b ) , then f 0 . Theorem 7.4.11. If f,g are analytic, then so are f ± g , f · g , f/g (for g 6 = 0 ), and f g (for dom f Im g ) § 7.4 Exercise: # Recommended: # 1.

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120 Math 413 Sequences and Series of Functions 7.5 Approximation by polynomials 7.5.1 Convolution and approximate identities Deﬁnition 7.5.1. The support of f is the closure of the largest open set on which f 6 = 0. Since the zero set of a function is closed, spt f = { x . . . f ( x ) = 0 } C . f is always deﬁned on all of spt f . The support of f is where does anything interesting. For example: Theorem 7.5.2. For f ∈ R ( D ) , R D f ( x ) dx = R spt f f ( x ) dx . Deﬁnition 7.5.3. If f,g ∈ R ( R ), then their convolution f * g is deﬁned by ( f * g )( x ) := Z f ( x - y ) g ( y ) dy. Convolution is a weighted average of translates. NOTE: if at least one of f,g has compact support, then the convolution will exist. For
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Unformatted text preview: the rest of this section, we assume that functions have compact support, or equivalently, that we are working on a compact interval. Convolution as a product Theorem 7.5.4. Suppose f,g,h ∈ R ( D ) , D compact. (i) (Linearity) f * ( g + h ) = ( f * g ) + ( f * h ) and ( cf ) * g = c ( f * g ) = f * ( cg ) , ∀ c ∈ C . Proof. Immediate from linearity of the integral. (ii) (Commutativity) f * g = g * f . Proof. Change of variables: y 7→ x-y . (iii) (Associativity) f * ( g * h ) = ( f * g ) * h . Proof. Fubini theorem: R R ϕ ( x,y ) dxdy = R R ϕ ( x,y ) dy dx . (iv) (Fourier transform) [ f * g ( ξ ) = ˆ f ( ξ )ˆ g ( ξ ) ....
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