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Distributions: How to Think About the Dirac Delta Function

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Distributions: How to Think About the Dirac Delta Function 1 Distributions on R Definition 1. Let X be a topological space, let f : X ! C, then define the support of f to be the set supp(f) = fx 2 X : f(x) 6= 0g: Definition 2. Let ΩRn, then define D(Ω) ½ C1(Ω) to be the space of infinitely differentiable (partials of all orders) functions f : Rn ! R which have compact support. We can see that D(Ω) is a real vector space, an algebra under pointwise multiplication, and an ideal in C1(Ω). We can always thing of D(Ω) ½ D(Rn), and for brevity I will call D(R) = D. n n Definition 3 (Convergence in D(R )). Let f'ig be a sequence in D(R ) n and let ' 2 D(R ), then we say that f'ig ! ' if n 1. supp('i) ½ K for some compact K ½ R for all i 2 N. p p n 2. fD 'ig ! D ' uniformly for each p 2 N . This topology defined by some semi-norm? n n Theorem 4. The space D(R ) is dense in C0(R ), the space of continuous real function of compact support with topology given by uniform conver- n n gence, i.e. for any f 2 C0(R ) with supp(f) ½ U for some open U ½ R and for any " > 0, there exists a ' 2 D(Rn) with supp(') ½ U, and such that for all x 2 Rn jf(x) ¡ '(x)j < ": Definition 5. A distribution T is a continuous linear map T : D(Rn) ! C, i.e. 1. T (®' + ¯Ã) = ®T (') + ¯T (Ã) for all ®; ¯ 2 C, '; à 2 D. 2. f'ig ! Á in D ) fT ('i)g ! T (') in C. The set of all distributions on Rn will be denoted D0(Rn). We can see that D0(Rn) ½ D¤(Rn), and assuming the axiom of choice one can show that there exist linear maps T : D(Rn) ! C which are not continuous in the topology of D(Rn). 1 Definition 6. Let L(Rn) be the space of all locally integrable functions f : Rn ! C, i.e. for all compact sets U ½ Rn, f is integrable on U. n 0 n Example 7. For every f 2 L(R ), we can define a distribution Tf 2 D (R ) given by Z Tf (') = f'; Rn for all ' 2 D(Rn), where the integral is the Lebesgue integral, and where really, the integral is finite since the support of ', hence of f', is bounded. n For any f 2 L(R ), the map Tf is obviously linear from the properties of the integral, now let f'ig ! ' in D, with supp('i) ½ K, then Z µZ ¶ jTf ('i) ¡ Tf (')j · jf('i ¡ ')j · jfj sup j'i(x) ¡ '(x)j ! 0: Rn K x2K So in fact Tf ('i) ! Tf ('), so Tf is continuous. Claim 8. Let f; g 2 L, then Tf = Tg , f = g almost everywhere. So there is an injection e 0 L!D ; f 7! Tf ; where Le = L=ff 2 L : f = 0 almost everywhereg. So we can think of locally integrable functions as distributions. There are distributions which do not accord to locally integrable functions. Example 9. The Dirac delta distribution ± 2 D0 is given by, ±(') = '(0); for all ' 2 D0: n Also for each a 2 R we can define ±a in the obvious way. Then obviously ± forms a distribution. In fact there exists no f 2 L such that ± = Tf , i.e. such that Z f(x)'(x)dx = '(0); for all ' 2 D: Rn 2 2 Derivatives of Distributions We want to develop the notion of the derivative of a distribution, so, we start by looking at derivative of derivatives corresponding to C1 functions. motivation Let f 2 C1(R), then for all ' 2 D, Z 0 Tf 0 (') = f (x)'(x)dx R ¯ Z ¯1 = f(x)'(x)¯ ¡ (f(x)'0(x)dx ¡1 Z R 0 0 = ¡ (f(x)' (x)dx = ¡Tf (' ): R This provides the motivation for defining the derivative of an arbitrary dis- tribution T 2 D0 by DT (') = ¡T (D'); for all ' 2 D; and for each k 2 N, DkT (') = (¡1)kT (Dk'); for all ' 2 D: We can see how this generalizes to Rn, for all p 2 Nn, DpT (') = (¡1)jpjT (Dp'); for all ' 2 D; where à ! à ! p1 pn p @ @ D = ¢ ¢ ¢ and jpj = p1 + ¢ ¢ ¢ pn @x1 @xn 1 Note that for any f 2 C , DTf = TDf . Example 10. Let the Heavyside function H : R ! R be given by ( 0 for x < 0 H(x) = : 1 for x ¸ 0 Then for any ' 2 D, Z 0 0 DTH (') = ¡TH (' ) = ¡ H(x)' (x)dx R Z ¯ 1 ¯1 = ¡ '0(x)dx = ¡'(x)¯ = '(0) = ±('); 0 0 3 thus we have found out that as distributions DH = ±. Now we want to see what D± turns out to be, for any ' 2 D, D±(') = ¡±('0) = ¡'0(0); and so in general, Dk±(') = (¡1)k'(k)(0): Example 11. Though the function f(x) = 1=x is not locally integrable, the distribution given by, for every ' 2 D, µ ¶ µ ¶ Z 1 Z ¡" Z 1 1 T (') = pv '(x)dx = lim + '(x)dx ; R x "!0 ¡1 " x does make sense. Let ' 2 D, and suppose supp(') ½ (¡a; a), then 3 Multiplication In general there is no way of multiplying two arbitrary distribution, this is because, even in the case of L, the product of locally integrable functionq is not necessarily locally integrable, an example is the function f(x) = 1= jxj which is locally integrable, however, f 2(x) = 1=jxj is not locally integrable. But we can multiply distributions by infinitely differentiable functions. First, for any f 2 L, ® 2 C1, ' 2 D, Z Z T®f (') = ®f' = f(®') = Tf (®'); Rn Rn since D is an ideal in C1. Using this as motivation, for any distribution T 2 D0, we define (®T )(') = T (®'); for all ' 2 D: Example 12. Let ® 2 C1, then for ' 2 D, (®±)(') = ±(®') = ®(0)'(0) = ®(0) ¢ ±('): In particular id± = 0: 4 Also we have, (®D±)(') = D±(®') = ¡(®')0(0) = ¡®(0)'0(0)¡®0(0)'(0) = (®(0)D±¡®0(0)±)('): In particular idD± = ¡±; id2D± = 0; idDk± = ¡kDk¡1±: Theorem 13 (Product rule for distributions). Let ® 2 C1, T 2 D0, then D(®T ) = ®(DT ) + (D®)T: Proof. Let ' 2 D, then D(®T )(') = ¡(®T )(D') = ¡T (®D') = ¡T (D(®') ¡ (D®)') = ¡T (D(®')) + T ((D®)') = (®(DT ))(') + (D®)T ('): 4 Convergence of Distributions Definition 14. A sequence of distributions fTig converges to T 2 D if fTi(')g ! T ('); for all ' 2 D: This is “pointwise convergence” or “weak” convergence of functionals. 0 Theorem 15. Let fTig be a sequence in D , then if for each ' 2 D, if 0 fTi(')g converges in C, then fTig ! T for some T 2 D . Theorem 16 (Dominated convergence). Let ffig be a sequence in L such that ffig ! f for some function f such that for all i 2 N, jfij · g for some g 2 L, then fTfi g ! Tf . Theorem 17. The derivative operator D : D0 !D0 is linear and continuous, i.e. if fTig ! T then fDTig ! DT . Proof. Suppose fTig ! T , then for all ' 2 D, fDTi(')g = f¡Ti(D')g ! ¡T (D') = DT ('): 5.
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