Question

This is a Fourier Analysis Question

TO SOLVE Exercise 27.4 (truncation) For fC(R), show that there exists φ E (R) that agrees with f on [-1, 1]. FOR REFERENCE, DO NOT SOLVE The basic idea for generalizing the notion of function in the context of distributions is to regard a function as an operator Ty (called a functional) acting by integration on functions themselves: and integration by parts shows that Ty(y) - 15.1.7 Definition (R) (or (I) will denote the space of functions in Co (R) (or C(I)) that have bounded support. 27.2 The space 9 (R) of test functions One is led naturally to require that test functions be infinitely differentiable and have bounded support. The space of these functions is denoted by (R) or simply (recall Definition 15.1.7) (i) φ vanishes outside a bounded interval (which depends on φ) (11) ф is infinitely differentiable in the usual sense on R. In other words, 9 (R)-(v : R Cl ф E Coo(R), supp(v) is bounded.) 27.3.1 Definition A distribution is any mapping that is linear and continuous. The variable of a distribution is a test function, and T(p) is a complex number. T is said to be a continuous linear functional (or linear form) on S. The value of Tat ф will be denoted in either of two ways: T(p) or (T,)

Answer 1

oC) that agree with -pont- Con ider te urction nte have c7o becaue the un t& +ve init supp . Now C. 米 whee

Cr 1:5 Note that idand o There-fae, thus show hau (X) -He he functm