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)