I. Introduction & Motivation I.1. Physical motivation. From [St]: consider a function T (x) as repre- senting the value of a physical variable at a particular point x in space. Is this a realistic thing to do? What can you measure? Suppose T (x) represents temperature at a point x in a room. You can measure the temperature with a thermometer, placing the bulb at the point x. Unlike the point, the bulb has nonzero size, so what you actually measure is more an average temperature over a small region of space. So really, the thermometer measures T (x)'(x) dx; Z where '(x) depends on the nature of the thermometer and where you place it. '(x) will tend to be \concentrated" near the location of the thermometer bulb and nearly 0 once you are sufficiently far away from the bulb. To say this is an average requires '(x) 0 for all x ≥ '(x) dx = 1 Z So it may be more meaningful to discuss things like T (x)'(x) dx than things like the value of T at a particular point x. R I.2. Mathematical motivation. How to “differentiate" nondifferentiable functions? IBP: b b T 0(x)'(x) dx = T (x)'0(x) dx; − Za Za (*) provided ' is nice, and the boundary terms vanish. Heaviside's operational calculus (c. 1900) ! Sobolev (1930s): realized generalized functions were continuous linear functionals! over some space of test functions Schwartz (1950s): developed & articulated this view in the language of mo! dern fnl analysis with Th´eorie des Distributions 1 Consider the DE u0 = H; where H(x) is the Heaviside function: 1; x > 0 H(x) = 0; x 0 ( ≤ H(x) Figure 1. The Heaviside function H(x). We would like to say that the soln is x; x > 0 x+ = 0; x 0; ( ≤ but this function is not a classical soln: it is not differential at 0. x+ Figure 2. The piecewise continuous function x+. Further, what is the derivative of H? Classical theory has no satisfac- tory treatment of derivatives of pointwise constant functions. (e.g. Cantor- Lebesgue). We will use IBP to differentiate the nondifferentiable. 2 II. Basics of the theory Let us not require T to have a specific value at x. T is no longer a function| call it a generalized function. Think of T as acting on a \weighted open set" instead of on a point x. \T (x) T (')" Formalize: T 7!acts on \test functions" '. Denote: T; ' = T (x)'(x) dx. h i Z ϕ(x) a b [ ] Figure 3. A test function '. Reqs for \nice" test functions ' : (1) ' C1, and (2) boundary2 terms must vanish ('( ) = '( ) = 0). −∞ 1 (a) ' Cc, i.e., ' has compact support, or 2 x (b) '(x) j j!1 0 quickly(with derivs) −−−−−! Choosing (2a) leads to the theory of distributions a` la Laurent Schwartz. Choosing (2b) leads to the theory of tempered distributions. Definition 1. The space of test functions is D (Ω) := Cc1(Ω) Note: (2a) allows more distributions than (2b). (larger class of test functions fewer distributions. ) Reqs for a distribution : (1) T; ' must exist for ' D. (2) Linearith i y: T; a ' + a 2' = a T; ' + a T; ' : h 1 1 2 2i 1h 1i 2h 2i Taking a cue from Riesz1: 1Actually, D0(Ω) is a larger class than the class of Borel measures on Ω. 3 Definition 2. The space of distributions is the (topological) dual space D0(Ω) = continuous linear functionals on D(Ω) : f g Examples 1. Any f L1 (Ω). 2 loc T ; ' = f(x)'(x) dx h f i Z \The distribution f" really means Tf . 2. Any regular Borel measure µ on Ω T ; ' = '(x) dµ h µ i Z T is regular iff T = T for f L1 (Ω). Otherwise, T is singular. f 2 loc The most famous singular distribution, Dirac-δ: δ; ' = '(0) = δ(x)'(x) dx = '(x) dµ h i so Z Z ; x = 0 \δ(x) = 1 " 0; x = 0 ( 6 for 1; 0 E µ(E) = 2 : 0; 0 = E ( 2 Some distributions do not even come from a measure. Example. δ0; ' := '0(0): h i − This defines a cont lin fnl on D (in fact, on C m; m 1), but not on C0.2 ≥ Note: require smooth test functions allow rough distributions; differen- tiability of a distribution relies on differen) tiability of test functions. To define δ; ' , ' must be C0. h i 1 To define δ0; ' , ' must be C . h i 2Riesz gives a bijection btwn contin lin fnls on C0 and complex locally finite regular Borel measures. 4 III. Differentiation of Distributions. Key point: how do we understand T ? Only by T; ' . h i So what does T 0 mean? Must understand T 0; ' . h i Working in Ω = R: 1 T 0; ' = T 0(x)'(x) dx h i Z−∞ 1 = [T (x)'(x)]1 T (x)'0(x) dx −∞ − Z−∞ = T; '0 −h i Definition 3. For any T D0(Ω), 2 D T; ' := T; D ' ; ' D: h k i −h k i 2 More generally (induct): α α α D T; ' := ( 1)j j T; D ' ; ' D: h i − h i 2 Note: this formula shows every distribution is (infinitely) differentiable. Note: Dα+βT = Dα(DβT ). Recall that x+ is not differentiable in the classical sense. x; x > 0 x+ = 0; x 0; ( ≤ But x+ can be differentiated as a distribution: x0 ; ' = x ; '0 h + i −h + i 1 = x'0(x) dx − Z0 1 = [ x'(x)]1 + '(x) dx − 0 Z0 1 = 0 + H(x)'(x) dx Z−∞ = H; ' h i So x+0 = H, as hoped. Similarly, 5 x00 ; ' = H0; ' h + i h i = H; '0 −h i 1 = '0(x) dx − Z0 = '(0) So x+00 = H0 = δ, as hoped. (Motivation for ' C1.) Note: it doesn't matter how rough T is | simply apply IBP until all deriv2 atives are on '. 6 IV. A Toolbox for Distribution Theory Definition 4. For ' 1 D, ' 0 iff f kgk=1 ⊆ k ! (i) K Ω compact s.t. spt ' K; k, and 9 ⊆ k ⊆ 8 (ii) Dα' 0 uniformly on K, α. k ! 8 Definition 5. For T 1 D0, f kgk=1 ⊆ T 0 T ; ' 0 C; ' D: k ! () h k i ! 2 8 2 This is weak or distributional (or \pointwise") convergence. Theorem 6. Differentiation is linear & continuous. Proof. a) (aT + bT )0; ' = aT + bT ; '0 h 1 2 i −h 1 2 i = a T ; '0 b T ; '0 − h 1 i − h 2 i = a T 0; ' + b T 0; ' h 1 i h 2 i b) Let T D0 converge to T in D0. f ng ⊆ Tn0 ; ' = Tn; '0 h n i − h i !1 T; '0 −−−−! − h i = T 0; ' h i Theorem 7. D0 is complete. Theorem 8. D is dense in D0. (Given T D0, ' D such that ' T as distributions.) 2 9f kg ⊆ k ! Proposition 9. If f has classical derivative f 0 and f 0 is integrable on Ω, then 0 Tf0 = Tf : 7 Proof. In the case Ω = I = (a; b), we have x f(x) = f 0(t) dt + f(c) x; c I: 2 Zc For ' D, (f')0 = f 0' + f'0: 2 Also, I (f')0 dx = 0 because f' vanishes outside a closed subinterval of I. So R f 0' + f'0 = 0: I I Hence Z Z T 0 ; ' = T ; '0 h f i −h f i = f'0 − ZI = f 0' ZI = T 0 ; ' h f i D0 1 ae 1 Theorem 10. fk Lloc, fk f, and fk g Lloc. Then fk f. D0 f g ⊆ −−! j j ≤ 2 −−−! (i.e., T T ) fk −−−! f D0 Definition 11. A sequence of functions fk such that fk δ is a delta- convergent sequence , or δ-sequence. f g −−−! Theorem 12. Let f 0 be integrable on Rn with f = 1. Define ≥ n x n x1 xn fλ(x) = λ− f λ = λ− f λ ; : : R: ; λ D0 for λ > 0. Then f δ as λ 0. λ −−−! ! Example. dx 1 = π = define f(x) = : R 1 + x2 ) π(1 + x2) Z Then obtain the delta-sequence 1 1 λ f = 2 = 2 2 : λ λ · π(1+(x/λ) ) π(x +λ ) 8 Example. 2 x2 1=2 e−x e− = π = define f(x) = pπ : R ) Z 2 −x /λ Then obtain the δ-sequence f = e : Further, λ pπλ n x 2 x2 e−| j dx = e− k dxk Rn Rn Z Z k=1 n Y x2 = e− k dxk Rn Yk=1 Z = πn=2 gives the n-dimensional δ-sequence 2 e x /λ f (x) = −| j : (1) λ (πλ)n=2 IV.1. Extension from test functions to distributions. There are two main ways to extend operations on test functions to opera- tions on distributions: via approximation or via adjoint identity. Approximation: For T D0, and an operator S defined on D, find arbitrary ' T and define ST =2lim S' . n ! n Example. (Translation) Define S = τh on D by τh(') = τh; ' = '(x h). h i − D0 To define τhT : find an arb sequence 'n D, with 'n T . Then T; ' = T (x)'(x) dx = lim f' (gx⊆)'(x) dx, so −−−! h i n R R 9 τ T; ' = lim τ ' (x)'(x) dx h h i h n Z = lim ' (x h)'(x) dx n − Z = lim 'n(x)'(x + h) dx Z = lim 'n(x)τ h'(x) dx − Z = T; τ h' h − i Example. (Differentiation) d D R Define S = dx on ( ).