Question

Exercise 27.1 Are the following functionals distributions? (a) T(p) Ip(0) (b) T(p)= а, а ЕС. Σ φ(n) (0). (c) T(p) n=0 27.2 The space (IR) of test funct i. One is led naturally to require that test functions he and have bounded support. The space of nitely 9 (R) or simply 9 (recall Definition 15.1.7), est functions y differentiable is denoted by of these functions vanishes outside a bounded interval (which depends on e). (İİ) ф is infinitely differentiable in the usual sense on R In other words, 9 (R)-(ф : R-C | ф E Coo(R), supp(p) is bounded.} mediately obvious how to construct such a function to ne dusual" functions (polynomials It is not im such functions exist, except for p 0. The "usual" functions (or rational functions, trigonometric functions, etc.) satisf In addition, the elementary infinitely differentiable functio (i) but not 6) ntable funetions are analytic. y (ii) Thus if they are zero on some interval, no matter how small halytio they lentically zero. This is almost the opposite of what we want. To f situation, we can try to define p a piece at a time: are x this θ(x) xe (a, b), if where θ is an elementary function. The difficulty here is that all of the derivatives of 0 must vanish to the left of a and to the right of b. There exists, however, at least one explicit example that is always presented in this context. It is the following function, which we have already seen in Lessons 15 and 21: exp(-if x

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