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By Walter E. Thirring

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5-11) Note that if qJ(x) is any test function, then IX(X)qJ(X) is also a test function, hence this definition makes sense. 6-4) (see Note below). O -+ 1(,), or even In(x) -+ I(x), but if the last form is used one must bear in mind that it is not pointwise convergence that is meant. As a rather trivial special case, if continuous functions In(x) converge uniformly to I(x), then as distributions they also converge to I(x). Mere pointwise convergence is not sufficient. Note. e. 4, hence is a distribution.

If the substitution x -+ a(x) is not one-to-one, or is not defined in alllR or is not onto all of IR, special definitions can often be made in such a way as to preserve formal rules of operation. For example, Dirac defines b(x 2 - a2 ) for a # Oby 2 2 b(x - a ) def 1 = 21al [b(x + a) + b(x - a)]. 8-7). 9 Restrictions, Limitations, and Warnings Distributions are in many ways like ordinary functions and are subject to most of the ordinary rules of operation. In fact, many of the limitations of those rules disappear in the context of distributions.

7-2) (1) First, if f(x) is an ordinary 1'(x), as h -+ 0. 7-3», but to prove this, the continuity of the fupctional (f, . 4 is needed. 7-3) holds if we can show that · \j, cp(x - h) - CP(X») = \j, I' cp(x - h) - CP(X») I1m, h ' 1m h . 7-5) h-O This equation follows from the continuity of the functional (j, . i -+ -III 'f' '( X ) , as h -+ 0. 7-6) See Exercise 1 below. 4, hence is a distribution. See Exercise 2 below. e. f(x + h) is the same distribution as f(x) for all h, it follows from the above that l' is the function 1'(x) == 0, hence f is the function f(x) = constant.

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