12.4. EXERCISES 315

4. ↑ Suppose f (x) = ∑∞k=1 uk (x) where the convergence is uniform and each uk is a

polynomial. Is it reasonable to conclude that f ′ (x) = ∑∞k=1 u′k (x)? The answer is no.

Use Problem 3 and the Weierstrass approximation theorem to show this.

5. Let X be a normed linear space. A ⊆ X is “weakly bounded” if for each x∗ ∈X ′, sup{|x∗(x)| : x ∈ A} < ∞, while A is bounded if sup{∥x∥ : x ∈ A} < ∞. ShowA is weakly bounded if and only if it is bounded.

6. ↑It turns out that the Fourier series sometimes converges to the function pointwise.Suppose f is 2π periodic and Holder continuous. That is | f (x)− f (y)| ≤ K |x− y|θwhere θ ∈ (0,1]. Show that if f is like this, then the Fourier series converges tof at every point. Next modify your argument to show that if at every point, x,| f (x+)− f (y)| ≤ K |x− y|θ for y close enough to x and larger than x and

| f (x−)− f (y)| ≤ K |x− y|θ

for every y close enough to x and smaller than x, then Sn f (x)→ f (x+)+ f (x−)2 , the

midpoint of the jump of the function. Hint: Use the Riemann Lebesgue lemma.

7. ↑ Let Y = { f such that f is continuous, defined on R, and 2π periodic}. Define∥ f∥Y = sup{| f (x)| : x ∈ [−π,π]}. Show that (Y,∥ ∥Y ) is a Banach space. Let x ∈ Rand define Ln( f ) = Sn f (x). Show Ln ∈ Y ′ but limn→∞ ∥Ln∥= ∞. Show that for eachx ∈ R, there exists a dense Gδ subset of Y such that for f in this set, |Sn f (x)| isunbounded. Finally, show there is a dense Gδ subset of Y having the property that|Sn f (x)| is unbounded on the rational numbers. Hint: To do the first part, let f (y)approximate sgn(Dn(x−y)). Here sgnr = 1 if r > 0,−1 if r < 0 and 0 if r = 0. Thisrules out one possibility of the uniform boundedness principle. After this, show thecountable intersection of dense Gδ sets must also be a dense Gδ set.

8. Let α ∈ (0,1]. Define, for X a compact subset of Rp,

Cα (X ;Rn)≡ {f ∈C (X ;Rn) : ρα (f)+∥f∥ ≡ ∥f∥α< ∞}

where ∥f∥ ≡ sup{|f(x)| : x ∈ X} and

ρα (f)≡ sup{ |f(x)− f(y)||x−y|α

: x,y ∈ X , x ̸= y}.

Show that (Cα (X ;Rn) ,∥·∥α) is a complete normed linear space. This is called a

Holder space. What would this space consist of if α > 1?

9. ↑Let X be the Holder functions which are periodic of period 2π . Define Ln f (x) =Sn f (x) where Ln : X →Y for Y given in Problem 7. Show ∥Ln∥ is bounded indepen-dent of n. Conclude that Ln f → f in Y for all f ∈ X . In other words, for the Holdercontinuous and 2π periodic functions, the Fourier series converges to the functionuniformly. Hint: Ln f (x) is given by

Ln f (x) =∫

π

−π

Dn (y) f (x− y)dy