MA5238 Fourier Analysis Wayne Lawton Department of Mathematics S , Lecture 9. Tues, Fri 2,5 March 2010

Locally Convex TVS This family can be chosen to be finite iff is defined by a family of seminorms. orover the field space and then if complete it is a Banach space. is a normed This family can be chosen to be countable iff is metrizable and then if complete it is a Fréchet space. Examples: for nonempty open Fréchet space defined by the seminorms is a Fréchet space defined by seminorms

Space of Rapidly Decreasing Functions such that space of infinitely differentiable functions for all with Fréchet space defined by seminorms: space of tempered distributions

Inductive Limit TVS where or LF-space is defined on the space Fréchet spaces and each inclusion is an isomorphism. In the LF-space a convex subsetis open iff is an open susbet of Then is the same as the topology on(a) The topology on (b)is not metrizable, filters are used for convergence (c)is complete – every Cauchy filter converges (d) a linear functionis continuous iff each restrictionis continuous (e) eachis nowhere dense in is a nested sequence for all as a subspace of

Radon Measures Open complex-valued continuous functions with compact support with the LF-topology the linear dual is the space of Radon Measures

Distributions Open complex-valued compactly supported functions + infinitely differentiable with the LP-topology linear dual is the space of distributions m-order derivatives are continuous with the LP-topology complex-valued compactly supported linear dual is the space of order-m distributions

Compactly Supported Radon Measures Open therefore

Compactly Supported Distributions Open therefore Definition The order ofis the smallest integer for which an inequality of the type above is valid. Remark This formal concept of order differs the intuitive notions discussed on page 84 in Strichartz’s textbook

Structure of Distributions of Finite Order and order Theorem 24.4 (p. 259 of Treves TVS, Dist. & Kernels) has supportand then there exists a family of Radon If is open and measuresonsuch that and Proof Since the number of multi-indices such thatequalsthe map defined by is onto a closed subspace and the HBT  extends to so there exists

Structure of Distributions of Finite Order Radon measures such that Now construct a functionthat equals 1 on a and whose support is contained inThen and open

Structure of Radon Measures Theorem 24.5 (p. 262 of Treves TVS, Dist. & Kernels) onEvery Radon measure onis a finite sum If of derivatives of order is an open subset of of functions in and Proof pages in Treves. such thatthen there exists Exampleis a Radon measure supported on Ifequals 1 in a neighborhood of H is the Heaviside function then and

Assignment 9 Read pages in Chapter 6 Do problems on pages for Fri 5 March Do problems 1-10 on pages for Tues 9 March Read pages in Chapter 7