Distributions and their Applications in Physics by F. Constantinescu

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The Distribution Space £>' The set of continuous linear functionals on 3D is denoted by SE> · The elements of £>' are called distributions of ©' or simply distributions. 10). shall now give a few examples of distributions. 3) defines a continuous linear functional on î>: whenever y v -> ip in f), (f, Y V ) —► (ff f)· A functional in $>' which can be written in the form (3·3) is said to be regular* Functionals in ©', which are not regular, are said to be singular» Regular functionals will be studied in Chapter 8 in more detail.

Proof. According on © is continuous such that γ ν -+0. such that γν-» 0. ifc) = lim (ffaipj = 0 is a sequence in SD such that aip^->0 and f is continuous. 7) for each i p * ß ( B n ) f (a*,*) = (S,a(0) = a(0)(sfy>) = (a(o)S,y>) . 8) Operations on Distributions 45 « J). Theorem 6,2. Suppose that f t » ' and ^ is an infinitely differentiable function which is equal to 1 on an open set containing supp f· Then f = ^f. Proof. supp f. Let if e £ . (ι-τ)ν) = o . Since this is true for each y e £ , it follows that f = *jf.

1) if |x| > e , where k is such that J|xi< 1 p e 2) · V1 |aq ' Let G be a bounded open subset of B K c G c G c U such that f and denote the characteristic function of G by X · Write d(Kf B n - G ) = inf {|x-y| j x * K and y * B n - G } and define d(ÏÏ, B n -U) i n a s i m i l a r way: d(K, B n -G) p o s i t i v e and so we can choose t s a t i s f y i n g 0 < ε (x) = J n X(y+*)p t (y)dy B = j X (y+x)pjy)dy = ] |y|<€. p£(y)dy = 1 · |y| ε, then X(x+y) pt(y) = 0 for all y e B , so that supp ψ c {xfeB which is contained in U.