The second is the wish that if f is a reasonably nice function in E and φ n

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1 8 Sectio : Approximatios i Reproducig Kerel Hilbert Spaces I this sectio, we address two cocepts. Oe is the wish that if {E, <, >} is a ierproduct space of real valued fuctios o the iterval [,], the there should be a fuctio K from [,]x[,] to the real umbers, such that if e is a umber i [,], the g(x) = K(x,e) is i E, ad if f is also i E, the < f(x), K(x,e) > = f(e). The secod is the wish that if f is a reasoably ice fuctio i E ad φ, =.. 5, is a sequece of fuctios which have graphs that are triagles with peaks at t p = p/5, p=..5, (See the ext graphs.) the there should be a reasoably good approximatio for f(x) as 5 f(x) ~ f( tp ) φ p ( x ). p = I some circumstaces these wishes come true. Both these wishes are related to otios cocerig use of the Dirac Delta fuctio. The first idea is related to the ofte used itegral f( x ) δ ( x, e) dx = f(e) where δ ( x, e ) is the Dirac Delta. The secod is the commo otio that reasoably ice fuctios ca be writte as sums of multiples of the Dirac Delta. We will show that the commoly used Dirac Delta caot exist i the Hilbert Space of square itegrable fuctios o [,]. The formulatio of such a fuctio is most ofte properly put i the cotext of distributios. Defiitio: A Hilbert Space of fuctios o [,] has a reproducig kerel if there is a fuctio K such that, for each e i [,], the fuctio g(x) = K(x,e) is i E ad, for each f i E, f(e) = < f(x), K(x,e) >.

2 8 Theorem. If { E, <, >} is a Hilbert Space of fuctios o [,] which has a reproducig kerel, the ormed covergece i E implies poitwise covergece o [,]. Moreover, if there is a umber B such that K(x,x) < B, the ormed covergece i E implies uiform covergece o [, ]. Proof: A proof follows from this iequality: Suppose f ( x ), =,, 3,... is a sequece i E. The f ( e ) - f m ( e ) = < f f m, K(,e) > f f m K(, e) = f f m K ( e, e ). Commets:. It follows that i the Hilbert Space of square itegrable fuctios o [, ], if e is i [,], the g(x) = δ(x,e) is ot i E. This follows because we kow that there are sequeces of fuctios i that space which coverge i the orm of E, but do ot coverge poitwise o [,].. We will show later that the property of havig ormed covergece to imply poitwise covergece i a Hilbert Space of fuctios o [,] is also a sufficiet coditio to assure that the Hilbert Space has a reproducig kerel. Example: Let E cosist of all fuctios f o [,] that are cotiuous ad for which the derivative f ' is cotiuous except possibly for a fiite umber of jumps. Also, take f() to be zero for all fuctios i E. Defie the ier product by < f, g > = f '(x) g '(x) dx. Take {E, <, >} to be the Hilbert Space formed from the completio of the ierproduct space defied above. Defie the fuctio K by K(x,y) = mi(x,y). Suppose f has a cotiuous derivative. The < f(x), K(x,y) > = f dx = f(y) - f() = f(y). x y Cosequetly, this Hilbert Space has a reproducig kerel. It is the K as defied above. Observatio : With K as defied for this example, if a < b s < t, the < K(x,b) - K(x,a), K(x,t) - K(x,s) > =. To see this, ote that if c < d, the the derivative of K(x,d) - K(x,c) is the characteristic fuctio o [c, d]. That is, it is the fuctio that is o [c, d] ad zero elsewhere. Thus, i the space of the example, the dot product of < K(x,b) - K(x,a), K(x,t) - K(x,s) > is zero for o overlappig itervals.

3 83 Observatio : If is a positive iteger, the the family { (K(x, p/) - K(x, (p-)/) }, p=,,..., is a orthoormal sequece i the Example above. To see that this observatio is true, the above Observatio establishes that the terms of the sequece are orthogoal. We have oly to see that they have orm. But, this is verified by itegratig the costat fuctio from (p-)/ to p/ ad multiply by. The result is. For illustratio, we draw the graph of oe of these fuctios i case = 5. Observatio 3: Take φ p, p=,,..., to be the orthoormal sequece defied above ad suppose that f is i the Hilbert Space of the Example. We kow that the best approximatio for f with the φ p 's is give by f ~ φ p ad that the coefficiets are the Fourier coefficiets: = < f, φ p >. Further, we kow how good the approximatio is f - φ p f -. Observatio 4: We calculate both terms of the right had side of the above iequality i case f ' is cotiuous except for a fiite umber f jumps: ad f = f '(x) dx,

4 84 p = 5 p f '(x) dx. Observatio 5: Recall that if f ' is cotiuous o the iterval [(p-)/, p/ ] the = f(p/) - f((p-)/) so that, i this case, we have a easy way to calculate the coefficiets. Two Approximatio Examples: Example. Suppose that f(x) = x o [,]. We compute (a) two, ad (d) draw a graph to illustrate the approximatio. f, (b), (c) the differece i these Here is the computatio for (a) Here is the computatio for (b) We ca make the computatio of (c): ad f = 4/3, f - 4 = 3 φ p 3, 3. f x x d dx ( x ) p p 4 dx = 3 d d = dx ( x ) x p ( 3 / ) ( p ) + 4 = 3 3 For (d) we draw the graphs with = 5 to show the closeess of the approximatio. The graph of f is black ad the approximatio is red.

5 85 approx x 5 c 3 5 c 5 c c c 5 5 K x, p K x, p Example. Suppose that f(x) = x ( x ) o [,]. We make the same computatios. Here is the computatio for (a) Here is the computatio for (b) d d = dx ( x ( x) ) x 3

6 86 We ca make the computatio of (c): ad f = /3, f - = 3 φ p 3, 3. p p d d = dx ( x ( x) ) x p + 3 / ( p + ) 3 = 3 ( ) For (d) we draw the graphs with = 5 to show the closeess of the approximatio. The graph of f is black ad the approximatio is red. approx x 4 5 c 5 c c 3 c 4 5 c K x, p K x, p

7 87 Alterate Computatios. We preset here computatios for the approximatio that are simpler ad make a iterestig geometric costructio. Here is the idea. Suppose we have a fuctio h which has a cotiuous derivative. The the Fourier Coefficiets are easily computed: h '(x) dx = h(b) - h(a). a b I this case, we ca write the approximatio as h p h p K x, p K x, p Observe that h(p/) is multiplied by ( K(x,p/) - K(x,(p-)/) - K(x,(p+)/)). Hece, a alterate represetatio for the approximatio that ivolves o computatio of coefficiets ad the associate itegratio ca be created as follows: defie T p ( x ) = ( K(x,p/) - K(x,(p-)/) - K(x,(p+)/)), p=,,.... The h(x) ~ h p T p ( x) We preset two illustratios. Example 3: Let h(x) = si( π x). We draw the graphs usig the approximatio for h give above ad with =. approx x h p T p ( x)

8 88 Example 4: I this example, we make a graph of this modified represetatio for the approximatio eve whe h'(x) is ot cotiuous. I this example, h is cotiuous, with a break i the derivative at x = /4. We use = 5 so that there is ot a ode poit at the break. approx x h p T p ( x) I this example, it is iterestig to see what are the supportig fuctios for the approximatio.

9 89 Fially, we preset the mai theorem of this sectio. Theorem. If { E, <, >} is a Hilbert Space of umber valued fuctios o [,]. These are equivalet: () There is a reproducig kerel for {E, <, >}. () If x is i [, ] ad L x is the fuctio from E to the real umber defied by L x (f) = f(x) for each f i E, the L x is cotiuous o E. (3) Normed covergece i E implies poitwise covergece o [, ]. Idicatio of proof: It is routie to show that () ad (3) are equivalet. We saw above that () implies (3). To see that () implies (), we use the Riesz Represetatio Theorem as follows. Suppose that x is i [, ] ad L x is defied as i (). To assert that this is a cotiuous fuctio from E to the umbers implies that there is h i E such that L x (f) = < f, h >. Clearly, this Riesz poit chages with x. Thus, we write, for each f i E, f(x) = L x (f) = < f, h x > = < f(t), h(t,x) >.