12.7. EXERCISES 291

11. Suppose f is a differentiable function of period 2π and suppose that both f and f ′

are in R([−π,π]) such that for all x ∈ (−π,π) and y sufficiently small,

f (x+ y)− f (x) =∫ x+y

xf ′ (t)dt.

Show that the Fourier series of f converges uniformly to f . Hint: First show usingthe Dini criterion that Sn f (x)→ f (x) for all x. Next let ∑

∞k=−∞

akeikx be the Fourierseries for f . Then from the definition of ak, show that for k ̸= 0,ak =

1ik a′k where

a′k is the Fourier coefficient of f ′. Now use the Bessel’s inequality to argue that

∑∞k=−∞

∣∣a′k∣∣2 < ∞ and then show this implies ∑ |ak| < ∞. You might want to usethe Cauchy Schwarz inequality in Theorem 2.15.1 to do this part. Then using theversion of the Weierstrass M test given in Problem 10 obtain uniform convergenceof the Fourier series to f .

12. Let f be a function defined on R. Then f is even if f (θ) = f (−θ) for all θ ∈ R.Also f is called odd if for all θ ∈ R, − f (θ) = f (−θ). Now using the Weier-strass approximation theorem show directly that if h is a continuous even 2π periodicfunction, then for every ε > 0 there exists an m and constants, a0, · · · ,am such that∣∣h(θ)−∑

mk=0 ak cosk (θ)

∣∣ < ε for all θ ∈ R. Hint: Note the function arccos is con-tinuous and maps [−1,1] onto [0,π] . Using this show you can define g a continuousfunction on [−1,1] by g(cosθ) = h(θ) for θ on [0,π]. Now use the Weierstrassapproximation theorem on [−1,1].

13. Show that if f is any odd 2π periodic function, then its Fourier series can be simpli-fied to an expression of the form ∑

∞n=1 bn sin(nx) and also f (mπ) = 0 for all m ∈ N.

14. Consider the symbol ∑∞k=1 an. The infinite sum might not converge. Summability

methods are systematic ways of assigning a number to such a symbol. The nth

Ceasaro mean σn is defined as the average of the first n partial sums of the se-ries. Thus σn ≡ 1

n ∑nk=1 Sk where Sk ≡ ∑

kj=1 a j. Show that if ∑

∞k=1 an converges then

limn→∞ σn also exists and equals the same thing. Next find an example where, al-though ∑

∞k=1 an fails to converge, limn→∞ σn does exist. This summability method is

called Ceasaro summability. Recall the Fejer means were obtained in just this way.

15. Modify Theorem 12.5.3 to consider the case of a piecewise continuous function f .Show that at every x, σn+1 ( f )(x)→ f (x+)− f (x−)

2 . This requires no extra conditions.Piecewise continuous is enough.

16. Let 0 < r < 1 and for f a periodic function of period 2π where f ∈ R([−π,π]) ,consider Ar f (θ) ≡ ∑

∞k=−∞

r|k|akeikθ where the ak are the Fourier coefficients of f .Show that if f is continuous, then limr→1−Ar f (θ) = f (θ) . Hint: You need to finda kernel and write as the integral of the kernel convolved with f . Then considerproperties of this kernel as was done with the Fejer kernel. In carrying out the details,you need to verify the convergence of the series is uniform in some sense in order toswitch the sum with an integral.

17. In the above problem, if f is piecewise continuous, can you show that convergencehappens to the midpoint of the jump?