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# K theorem 105 schwartz representation any tempered

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Unformatted text preview: sing its completeness). However u = u� as before, so u → C0 (Rn ). � In particular we see that (10.6) H � (Rn ) = � H m (Rn ) � C � (Rn ) . m These functions are not in general Schwartz test functions. Proposition 10.4. Schwartz space can be written in terms of weighted Sobolev spaces � (10.7) S (Rn ) = ⊂x≤−k H k (Rn ) . k LECTURE NOTES FOR 18.155, FALL 2004 65 Proof. This follows directly from (10.5) since the left side is contained in � k ⊂x≤−k C0 −n (Rn ) � S (Rn ). k � Theorem 10.5 (Schwartz representation). Any tempered distribution can be written in the form of a ﬁnite sum � � 0 (10.8) u = x� Dx u�� , u�� → C0 (Rn ). |�|�m |� |�m or in the form (10.9) u= � � 0 Dx (x� v�� ), v�� → C0 (Rn ). |�|�m |� |�m Thus every tempered distribution is a ﬁnite sum of derivatives of continuous functions of p oynomial growth. Proof. Essentially by deﬁnition any u → S � (Rn ) is continuous with re­ spect to one of the norms ∈⊂x≤k α∈C k . From the...
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## This note was uploaded on 01/22/2014 for the course MATH 18.155 taught by Professor Melrose during the Fall '13 term at MIT.

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