convolution - 14 1 The Fourier Transform on L 1 ( R ) 1.8....

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Unformatted text preview: 14 1 The Fourier Transform on L 1 ( R ) 1.8. This problem provides an alternative proof to Theorem 1.17. (a) Show that hatwide f C ( R ) for every f S = span { [ a,b ] : a < b R } . (b) Show that S is dense in L 1 ( R ) (see Exercise B.61), and use this to prove that hatwide f C ( R ) for every f L 1 ( R ). 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 pointwise 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 multiplication-like operation under which L 1 ( R ) is closed. This operation, convolution of functions, will be one of the most important tools in our further development of harmonic analysis. Therefore, in this section we set aside the Fourier transform for the moment, and concentrate on developing the machinery of convolution. 1.3.1 Some Notational Conventions Before proceeding, there are some technical issues related to the definition of elements of L p ( R ) that we need to clarify (see Section B.6.2 for additional discussion of these issues). The basic source of difficulty is that an element f of L p ( R ) is not a func- tion but rather denotes an equivalence class of functions that are equal almost everywhere. Therefore we cannot speak of the value of f L p ( R ) at a point x R , and consequently concepts such as continuity or support do not ap- ply in a literal sense to elements of L p ( R ). For example, the zero function 0 and the function Q both belong to the zero element of L p ( R ), which is the equivalence class of functions that are zero a.e., yet 0 is continuous and com- pactly supported while Q is discontinuous and its support is R . Even so, it is often essential to consider smoothness or support properties of functions, and we therefore adopt the following conventions when discussing the smoothness or the support of elements of L p ( R ). More generally, these same issues and conventions apply to elements of L 1 loc ( R ) = braceleftbig f : R C : f K L 1 ( R ) for every compact K R bracerightbig , which is the space of locally integrable functions on R . Note that L p ( R ) L 1 loc ( R ) for every 1 p . 1.3 Convolution 15 Notation 1.19 (Continuity for Elements of L 1 loc ( R ) ). We will say that f L 1 loc ( R ) is continuous if there is a representative of f that is continuous, i.e., there exists some continuous function f such that f is the equivalence class of all functions that equal f almost everywhere....
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convolution - 14 1 The Fourier Transform on L 1 ( R ) 1.8....

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