420 CHAPTER 17. INNER PRODUCT SPACES

This is called the Dirichlet kernel. Show that

Dn (t) =1

2π

sin(n+(1/2)) tsin(t/2)

For V the vector space of piecewise continuous functions, define Sn : V 7→V by

Sn f (x) =∫

π

−π

f (y)Dn (x− y)dy.

Show that Sn is a linear transformation. (In fact, Sn f is not just piecewise continuousbut infinitely differentiable. Why?) Explain why

∫π

−πDn (t)dt = 1. Hint: To obtain

the formula, do the following.

ei(t/2)Dn (t) =1

2π

n

∑k=−n

ei(k+(1/2))t

ei(−t/2)Dn (t) =1

2π

n

∑k=−n

ei(k−(1/2))t

Change the variable of summation in the bottom sum and then subtract and solve forDn (t).

17. ↑Let V be an inner product space and let U be a finite dimensional subspace with anorthonormal basis {ui}n

i=1. If y ∈V, show

|y|2 ≥n

∑k=1|⟨y,uk⟩|

2

Now suppose that {uk}∞

k=1 is an orthonormal set of vectors of V . Explain why

limk→∞

⟨y,uk⟩= 0.

When applied to functions, this is a special case of the Riemann Lebesgue lemma.

18. ↑Let f be any piecewise continuous function which is bounded on [−π,π] . Show,using the above problem, that

limn→∞

∫π

−π

f (t)sin(nt)dt = limn→∞

∫π

−π

f (t)cos(nt)dt = 0

19. ↑∗Let f be a function which is defined on (−π,π]. The 2π periodic extension is givenby the formula f (x+2π) = f (x) . In the rest of this problem, f will refer to this 2π

periodic extension. Assume that f is piecewise continuous, bounded, and also thatthe following limits exist

limy→0+

f (x+ y)− f (x+)

y, lim

y→0+

f (x− y)− f (x+)

y

Here it is assumed that

f (x+)≡ limh→0+

f (x+h) , f (x−)≡ limh→0+

f (x−h)