140 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES

[0,1/2] ,−x ≥ ln(1− x) ≥ −2x. See [8] which is where I read this. That product is

∏ j≤n

(1−(

1− |m−p j|(p j+m+1)

))and so ln of this expression is

n

∑j=1

ln

(1−

(1−

∣∣m− p j∣∣

(p j +m+1)

))

which is in the interval[−2

n

∑j=1

(1−

∣∣m− p j∣∣

(p j +m+1)

),−

n

∑j=1

(1−

∣∣m− p j∣∣

(p j +m+1)

)]

and so dn → 0 if and only if ∑∞j=1

(1− |m−p j|

(p j+m+1)

)= ∞. Since pn → ∞ it suffices

to consider the convergence of ∑ j

(1− p j−m

(p j+m+1)

)= ∑ j

(2m+1

(p j+m+1)

). Now recall

theorems from calculus.

16. For f ∈ C ([a,b] ;R) , real valued continuous functions, let | f | ≡(∫ b

a | f (t)|2)1/2

≡

( f , f )1/2 where ( f ,g) ≡∫ b

a f (x)g(x)dx. Recall the Cauchy Schwarz inequality|( f ,g)| ≤ | f | |g| . Now suppose 1

2 < p1 < p2 · · · where limk→∞ pk = ∞. Let Vn =span(1, fp1 , fp2 , ..., fpn) . For ∥·∥ the uniform approximation norm, show that for ev-ery g ∈C ([0,1]) , there exists there exists a sequence of functions, fn ∈Vn such that∥g− fn∥→ 0. This is the second Müntz theorem. Hint: Show that you can approxi-mate x→ xm uniformly. To do this, use the above Müntz to approximate mxm−1 with∑k ckxpk−1 in the inner product norm.

∫ 10

∣∣mxm−1−∑nk=1 ckxpk−1

∣∣2 dx ≤ ε2. Thenxm−∑

nk=1

ckpk

xpk =∫ x

0(mtm−1−∑

nk=1 ckt pk−1

)dt. Then∣∣∣∣∣xm−

n

∑k=1

ck

pkxpk

∣∣∣∣∣≤∫ x

0

∣∣∣∣∣mtm−1−n

∑k=1

ckt pk−1

∣∣∣∣∣dt ≤∫ 1

01

∣∣∣∣∣mtm−1−n

∑k=1

ckt pk−1

∣∣∣∣∣dt

Now use the Cauchy Schwarz inequality on that last integral to obtain

maxx∈[0,1]

∣∣∣∣∣xm−n

∑k=1

ck

pkxpk

∣∣∣∣∣≤ ε.

In case m = 0, there is nothing to show because 1 is in Vn. Explain why the resultfollows from this and the Weierstrass approximation theorem.

17. Suppose f : [a,b]→ [0,1] is piecewise linear, equal to 1 on [a+h,b−h] and 0 at a,b.Show that

∫ ba f (x)dx = h+(b−a−2h) = b−a−h.