19.4. THE MÜNTZ THEOREM 531

in whichyT

x ≡ ((y,x1) , · · · ,(y,xn)) , αT≡(α1, · · · ,αn) .

Then 19.3.22 and 19.3.23 imply the following system(G (x1, · · · ,xn) 0

yTx 1

)(α

d2

)=

(yx

||y||2)

By Cramer’s rule,

d2 =

det(

G (x1, · · · ,xn) yx

yTx ∥y∥2

)det(

G (x1, · · · ,xn) 0yT

x 1

)

=

det(

G (x1, · · · ,xn) yx

yTx ∥y∥2

)det(G (x1, · · · ,xn))

=det(G (x1, · · · ,xn,y))det(G (x1, · · · ,xn))

=G(x1, · · · ,xn,y)G(x1, · · · ,xn)

and this proves the theorem.

19.4 The Müntz TheoremRecall the polynomials are dense in C ([0,1]) . This is a consequence of the Weierstrassapproximation theorem. Now consider finite linear combinations of the functions, t pk

where {p0, p1, p2, · · ·} is a sequence of nonnegative real numbers, p0 ≡ 0. The Müntztheorem says this set, S of finite linear combinations is dense in C ([0,1]) exactly when∑

∞k=1

1pk

= ∞. There are two versions of this theorem, one for density of S in L2 (0,1) andone for C ([0,1]) . The presentation follows Cheney [33].

Recall the Cauchy identity presented earlier, Theorem 5.5.5 on Page 79 which is statedhere for convenience.

Theorem 19.4.1 The following identity holds.

∏i, j

(ai +b j)

∣∣∣∣∣∣∣1

a1+b1· · · 1

a1+bn...

...1

an+b1· · · 1

an+bn

∣∣∣∣∣∣∣= ∏j<i

(ai−a j)(bi−b j) . (19.4.25)

Lemma 19.4.2 Let m, p1, · · · , pn be distinct real numbers larger than −1/2. Thus thefunctions, fm (x)≡ xm, fp j (x)≡ xp j are all in L2 (0,1). Let

M = span( fp1 , · · · , fpn) .

Then the L2 distance, d between fm and M is

d =1√

2m+1

n

∏j=1

∣∣m− p j∣∣

m+ p j +1