Kenneth Kuttler

454 CHAPTER 16. APPROXIMATION OF FUNCTIONS AND THE INTEGRAL

show if m = 0. Thus, assume m > 0. From the above theorem, consider n large enough that|m fm−1−h| < ε for some h ∈ span( fp1−1, ..., fpn−1) ,h(x) = ∑

nk=1 ck fpk−1 (x) . Then note

that xm =∫ x

0 m fm−1 (t)dt and also H (x) ≡∫ x

0 h(t)dt ∈ Vn. Therefore, from the CauchySchwarz inequality,

|xm−H (x)| =

∣∣∣∣∫ x

0(m fm−1 (t)−h(t))dt

∣∣∣∣≤ ∫ 1

01 |m fm−1 (t)−h(t)|dt

≤ 1(∫ 1

0|m fm−1 (t)−h(t)|2 dt

)1/2

= |m fm−1−h|< ε

Since this is true for each x, it follows that ∥ fm−H∥ ≤ ε . Then the same argument usedabove, depending on the triangle inequality proves the theorem. ■

Note that, as before, this shows that if g is continuous, there is a sequence of hn con-sisting of linear combinations of the fpk which converges uniformly to g.

Example 16.5.3 Let pk ≡ ln(1+ k) . Then if g is continuous on [0,1] , there is a functionof the form c0 +∑

Lk=1 ckxln(1+k) ≡ h(x) such that ∥h−g∥< ε . You could replace ln(1+ k)

with k ln(1+ k) or 5k and draw the same conclusion.

16.6 Exercises1. Show the above argument used to obtain Theorems 16.5.1, 16.5.2 works to give both

of these theorems if your interest is in [−1,1] rather than [0,1] provided you replacex with |x| whenever xp occurs for p not an integer. You must do something like thisbecause if x < 0, maybe xp is not well defined in the context of real analysis.

2. Generalize Theorems 16.5.1, 16.5.2 to any interval [a,b].

3. Show that the same arguments will work for proving Theorems 16.5.1, 16.5.2 inapproximating functions in C ([0,1] ;X) where X is an inner product space if you de-fine a new inner product ( f ,g) ≡

∫ 10 ( f ,g)dx. Show how to use this to generalize

Theorems 16.5.1, 16.5.2 to the case of many variables as was done for the Weier-strass theorem. Then, using the ideas of the above problem, show how to considerapproximation of functions C (R;X) where R is some n dimensional box of the form∏

ni=1 [ai,bi]. How would you generalize to the case of C (R,R) where R is just some

closed and bounded set?

454 CHAPTER 16. APPROXIMATION OF FUNCTIONS AND THE INTEGRALshow if m = 0. Thus, assume m > 0. From the above theorem, consider n large enough that|mfn—1 —h| < € for some h € span (fp,—1, ++) fp,—1) A(x) = Ly ce fp,—1 (x). Then notethat x” = [5 mfm—1(t)dt and also H (x) = fj h(t) dt € V,. Therefore, from the CauchySchwarz inequality,1k"-H(x)| = < [tlmfna() h(tIA1 1/2i(/ inns (0) —h(0)P dt) = |mfn_1—h| <€0Since this is true for each x, it follows that || fm —H|| < €. Then the same argument usedabove, depending on the triangle inequality proves the theorem. MlNote that, as before, this shows that if g is continuous, there is a sequence of h, con-sisting of linear combinations of the f,, which converges uniformly to g.Example 16.5.3 Let p, = In(1+k). Then if g is continuous on [0,1], there is a functionof the form co +L, yx +) = h(x) such that ||h — g\| < €. You could replace \n(1 +k)with kin(1 +k) or 5k and draw the same conclusion.16.6 Exercises1. Show the above argument used to obtain Theorems 16.5.1, 16.5.2 works to give bothof these theorems if your interest is in [—1, 1] rather than [0,1] provided you replacex with |x| whenever x? occurs for p not an integer. You must do something like thisbecause if x < 0, maybe x? is not well defined in the context of real analysis.2. Generalize Theorems 16.5.1, 16.5.2 to any interval |a, b].3. Show that the same arguments will work for proving Theorems 16.5.1, 16.5.2 inapproximating functions in C ([0, 1];X) where X is an inner product space if you de-fine a new inner product (f,g) = fo (f,g) dx. Show how to use this to generalizeTheorems 16.5.1, 16.5.2 to the case of many variables as was done for the Weier-strass theorem. Then, using the ideas of the above problem, show how to considerapproximation of functions C (R;X) where R is some n dimensional box of the form”_, [ai,bi]. How would you generalize to the case of C (R,R) where R is just someclosed and bounded set?