570 CHAPTER 21. BANACH SPACES

6. Let f be a 2π periodic locally integrable function on R. The Fourier series for f isgiven by

∞

∑k=−∞

akeikx ≡ limn→∞

n

∑k=−n

akeikx ≡ limn→∞

Sn f (x)

whereak =

12π

∫π

−π

e−ikx f (x)dx.

ShowSn f (x) =

∫π

−π

Dn (x− y) f (y)dy

where

Dn(t) =sin((n+ 1

2 )t)2π sin( t

2 ).

Verify that∫

π

−πDn (t)dt = 1. Also show that if g ∈ L1 (R) , then

lima→∞

∫R

g(x)sin(ax)dx = 0.

This last is called the Riemann Lebesgue lemma. Hint: For the last part, assume firstthat g ∈C∞

c (R) and integrate by parts. Then exploit density of the set of functions inL1 (R).

7. ↑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 Problem 6.

8. ↑ 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.

9. 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}