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# approxid - 1 The Fourier Transform on L1(R 38 which are...

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38 1 The Fourier Transform on L 1 ( R ) which are dense subspaces of L p ( R ). On these domains, P : D P L p ( R ) and M : D M L p ( R ). Show, however, that P and M are unbounded even when restricted to these domains, i.e., sup f D P , bardbl f bardbl p =1 bardbl Pf bardbl p = = sup f D M , bardbl f bardbl p =1 bardbl Mf bardbl p . 1.5 Approximate Identities Although L 1 ( R ) is closed under convolution, we have seen that it has no identity element. In this section we will show that there are functions in L 1 ( R ) that are “almost” identity elements for convolution. We will construct families of functions { k λ } λ> 0 which have the property that f k λ converges to f in L 1 ( R ) (and in fact in many other senses, depending on what space f belongs to). If k λ is a “nice” function, then f k λ will inherit that niceness, and so we will have a nice function that is arbitrarily close to f . Using this procedure we will be able to show that many spaces of nice functions are dense in L 1 ( R ), including C c ( R ), C m c ( R ) for each m N , and even C c ( R ). 1.5.1 Definition and Existence of Approximate Identities The properties that a family { k λ } λ> 0 will need to possess in order to be an approximate identity for convolution are listed in the next definition. Definition 1.51. An approximate identity or a summability kernel is a family { k λ } λ> 0 of functions in L 1 ( R ) such that (a) integraldisplay k λ ( x ) dx = 1 for every λ > 0, (b) sup λ bardbl k λ bardbl 1 < , and (c) for every δ > 0, lim λ →∞ integraldisplay | x |≥ δ | k λ ( x ) | dx = 0 . Note that, by definition, an approximate identity is a family of integrable functions. If it is the case that k λ 0 for each λ , then requirement (b) in Definition 1.51 follows from requirement (a). However, in general the elements of an approximate identity need not be nonnegative functions. The “easy” way to create an approximate identity is through dilation of a single function. Exercise 1.52. Let k L 1 ( R ) be any function that satisfies integraldisplay k ( x ) dx = 1 .

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1.5 Approximate Identities 39 Define k λ by an L 1 -normalized dilation: k λ ( x ) = λ k ( λx ) , λ > 0 , and show that the resulting family { k λ } λ> 0 forms an approximate identity. Note that there is an inherent ambiguity in our notation: We may use { k λ } λ> 0 to indicate a generic family of functions indexed by λ , or, as intro- duced in Notation 1.5, we may use k λ to denote the L 1 -normalized dilation of a function k . The intended meaning is usually clear from context. In any case, if we define k λ by dilation, then, as λ increases, k λ becomes more and more similar to our intuition of what a “ δ -function” (a function that is an identity for convolution) would look like (see the illustration in Figure 1.7 and the related discussion in Section 1.3.5). While there is no such identity for convolution in L 1 ( R ), the collection of functions { k λ } λ> 0 in some sense forms an approximation to this nonexistent δ -function, for requirement (c) implies that k λ becomes more and more concentrated near the origin as λ increases, with integraltext k λ = 1 for every λ .
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approxid - 1 The Fourier Transform on L1(R 38 which are...

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