290 CHAPTER 12. SERIES AND TRANSFORMS
to an arbitrary interval. This is another approach to the Weierstrass approximationtheorem. Hint: First find a linear function ax+b = y such that f −y has the propertythat it has the same value at both ends of [−π,π]. Therefore, you may consider thisas the restriction to [−π,π] of a continuous periodic function F . Now find a trigpolynomial, σ (x) ≡ a0 +∑
nk=1 ak coskx+ bk sinkx such that ∥σ −F∥ < ε
3 . Recall12.4. Now consider the power series of the trig functions making use of the errorestimate for the remainder after m terms.
4. The inequality established above,
2π
n
∑k=−n|ak|2 ≤
∫π
−π
|Sn f (θ)|2 dθ ≤∫
π
−π
| f (θ)|2 dθ
is called Bessel’s inequality. Use this inequality to give an easy proof that for allf ∈ R([−π,π]) , limn→∞
∫π
−πf (x)einxdx = 0. Recall that in the Riemann Lebesgue
lemma | f | ∈ R((a,b]) so while this exercise is easier, it lacks the generality of theearlier proof.
5. Let f (x) = x for x ∈ (−π,π) and extend to make the resulting function defined onR and periodic of period 2π . Find the Fourier series of f . Verify the Fourier seriesconverges to the midpoint of the jump and use this series to find a nice formula forπ
4 . Hint: For the last part consider x = π
2 .
6. Let f (x) = x2 on (−π,π) and extend to form a 2π periodic function defined on R.Find the Fourier series of f . Now obtain a famous formula for π2
6 by letting x = π .
7. Let f (x) = cosx for x∈ (0,π) and define f (x)≡−cosx for x∈ (−π,0). Now extendthis function to make it 2π periodic. Find the Fourier series of f .
8. Suppose f ,g ∈ R([−π,π]). Show 12π
∫π
−πf gdx = ∑
∞k=−∞
αkβ k, where αk are theFourier coefficients of f and β k are the Fourier coefficients of g.
9. Suppose f (x) = ∑∞k=1 ak sinkx and that the convergence is uniform. Recall some-
thing like this holds for power series. Is it reasonable to suppose that f ′ (x) =∑
∞k=1 akk coskx? Explain.
10. Suppose |uk (x)| ≤ Kk for all x ∈ D where
∞
∑k=−∞
Kk = limn→∞
n
∑k=−n
Kk < ∞.
Show that ∑∞k=−∞
uk (x) converges converges uniformly on D in the sense that for allε > 0, there exists N such that whenever n > N,∣∣∣∣∣ ∞
∑k=−∞
uk (x)−n
∑k=−n
uk (x)
∣∣∣∣∣< ε
for all x ∈ D. This is called the Weierstrass M test. The earlier version only dealtwith sums in one direction.