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TmprDistii02

# TmprDistii02 - Math 8602 Spring 94 Some notes on tempered...

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Math 8602 Spring ‘94 Some notes on tempered distributions. 1 Assigning a topology to the Schwartz class The Schwartz class is interesting, and applicable, with the Fourier transform and the tools of Lebesgue measure. We will define open sets for S , and with this topology, introduce the space S of continuous linear functionals on S . These objects are also called tempered distributions . They “include” all the L p spaces, S itself, and much more besides. The key ideas are these: regarding the action of a linear functional as a “formal” integral, and the “transpose” of a linear mapping. We have seen this in two places recently: the equations f x k g dx = – f g x k dx for integration by parts, and ^ f( ξ ) g( ξ ) d ξ = f(x) ^ g(x) dx, that says we can “move the hat.” The first equation is valid with f in S and g in M . In that equation, the left side makes sense for any measurable function g bounded by a polynomial, if f is in S . So we define a linear functional on S by the equation T g (f) := f(x) g(x) dx. The left side of the first equation can then be expressed as T g ( f x k ). If g is in M, then we can write the equation as T g ( f x k ) = –T g/ x k (f) . The right side may not make sense, if g is not in M , so we define g x k to be the linear functional , defined on S , given by the left side. The same is true in the second equation. The second equation is valid with f in S and g in S . The left side makes sense if f is in S and g is in M . But the right side does not — how do we define the Fourier transform of |x| 2 , for example? The point is, we define it by the equation, not as a function, but as a linear functional . In fact, we have ^ f( ξ ) ξ k 2 = – ^ f( ξ ) (i ξ k ) 2 = – 2 f x k 2 ^ ( ξ ), so ^ f( ξ ) | ξ | 2 = – ( f)^( ξ ), where denotes the Laplace operator. Now the equation reads ( f) ^( ξ ) d ξ = (|x| 2 )^(f). The left side contains the factor 1 = e –i ξ• 0 . By the inversion theorem we recognize this as –(2 π ) n ( f)(0). Thus we will be able to say (|x| 2 )^ = –(2 π ) n ( •)(0). two possible topologies for S We have infinitely many norms on S , namely the || f || αβ , with α and β in N n . Each of these gives us a collection of open sets, i.e. a topology on S . There are two immediate possibilities: we can define our topology to be the intersection of all these norm topologies. This will certainly yield a topology. In fact, we can use this fact to talk about the smallest topology containing a given collection of sets. The other possibility is the one we will use: the smallest topology that contains all these norm topologies . It is larger than the first one. Thus it is “easier” for a function that has S as its domain to be continuous. Let X be a set that has a topology. Then, f: S X is continuous with respect to our first topology if and only if f is continuous with respect to every topology in the collection. On the other hand, f: S X is continuous with respect to our second topology if f is continuous with respect to just one topology in the collection.

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TmprDistii02 - Math 8602 Spring 94 Some notes on tempered...

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