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MA5238 Fourier Analysis

Lecture 9. Tues, Fri 2,5 March 2010. MA5238 Fourier Analysis. Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1. over the field. or. is defined by a family of seminorms.

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MA5238 Fourier Analysis

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  1. Lecture 9. Tues, Fri 2,5 March 2010 MA5238 Fourier Analysis Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

  2. over the field or is defined by a family of seminorms. This family can be chosen to be finite iff is a normed Locally Convex TVS space and then if complete it is a Banach space. 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

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

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

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

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

  7. Open Compactly Supported Radon Measures therefore

  8. Compactly Supported Distributions Open therefore Definition The order of is 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

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

  10. Structure of Distributions of Finite Order such that Radon measures Now construct a function that equals 1 on a open and whose support is contained in Then and

  11. Structure of Radon Measures Theorem 24.5 (p. 262 of Treves TVS, Dist. & Kernels) Every Radon measure on on is a finite sum of derivatives of order of functions in If is an open subset of and then there exists such that Proof pages 262-263 in Treves. Example is a Radon measure supported on If equals 1 in a neighborhood of and H is the Heaviside function then

  12. Read pages 91-108 in Chapter 6 Do problems 20-42 on pages 110-112 for Fri 5 March Assignment 9 Read pages 113-125 in Chapter 7 Do problems 1-10 on pages 157-158 for Tues 9 March

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