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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→ xy . (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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 Fall '11
 Wong
 Topology, Power Series, Mathematical analysis, Closed set, Proof. Fubini theorem

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