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Unformatted text preview: 1.3 Convolution 15 1.3 Convolution Since L 1 ( R ) is a Banach space, we know that it has many useful properties. In particular the operations of addition and scalar multiplication are continuous. However, there are many other operations on L 1 ( R ) that we could consider. One natural operation is multiplication of functions, but unfortunately L 1 ( R ) is not closed under multiplication. Exercise 1.18. Show that f , g L 1 ( R ) does not imply fg L 1 ( R ). In this section we will define a different operation that L 1 ( R ) is closed under. This operation, convolution of functions, will be one of our most im- portant tools for the further development of harmonic analysis. Therefore, in this section we set aside the Fourier transform for the moment, in order to develop the machinery of convolution. 1.3.1 Definition and Basic Properties of Convolution Definition 1.19 (Convolution). Let f : R C and g : R C be Lebesgue measurable functions. Then the convolution of f with g is the function f * g given by ( f * g )( x ) = Z f ( y ) g ( x- y ) dy, (1.6) whenever this integral is well-defined. For example, suppose 1 p and p is its dual index. If f L p ( R ) and g L p ( R ), then (as functions of y ), f ( y ) and g ( x- y ) belong to dual spaces, and hence by H olders Inequality the integral defining ( f * g )( x ) in equation (1.6) exists for every x , and furthermore is bounded as a function of x . Exercise 1.20. Show that if 1 p , f L p ( R ), and g L p ( R ), then f * g L ( R ), and we have k f * g k k f k p k g k p . (1.7) We will improve on this exercise (in several ways) below. In particular, Exercise 1.20 does not give the only hypotheses on f and g which imply that f * g exists we will shortly see Youngs Inequality , which is a powerful result that tells us that f * g will belong to a particular space L r ( R ) whenever f L p ( R ), g L q ( R ), and we have the proper relationship among p , q , and r (specifically, 1 p + 1 q = 1 + 1 r ). However, before turning to that general case, we prove the fundamental result that L 1 ( R ) is closed under convolution, and that the Fourier transform interchanges convolution with multiplication. Theorem 1.21. If f , g L 1 ( R ) are given, then the following statements hold. 16 1 The Fourier Transform on L 1 ( R ) (a) f ( y ) g ( x- y ) is Lebesgue measurable on R 2 . (b) For almost every x R , f ( y ) g ( x- y ) is a measurable and integrable function of y , and hence ( f * g )( x ) is defined for a.e. x R . (c) f * g L 1 ( R ) , and k f * g k 1 k f k 1 k g k 1 . (d) The Fourier transform of f * g is the product of the Fourier transforms of f and g : ( f * g ) ( ) = b f ( ) b g ( ) , R ....
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