- Suppose f has infinitely many derivatives and is also periodic with period 2π. Let the
Fourier series of f be
Show that

for every m ∈ ℕ.

- The proof of Theorem 11.6.5 used the inequality
^{2}≤ 4whenever a,b and c are nonnegative numbers. In fact the 4 can be replaced with 3. Show this is true. - Let f be a continuous function defined on . Show there exists a polynomial, p such that< ε where
Extend this result to an arbitrary interval. This is another approach to the Weierstrass approximation theorem. Hint: First find a linear function ax + b = y such that f − y has the property that it has the same value at both ends of

. Therefore, you may consider this as the restriction toof a continuous periodic function F. Now find a trig polynomial,such that

<. Recall 11.4. Now consider the power series of the trig functions making use of the error estimate for the remainder after m terms. - The inequality established above,
is called Bessel’s inequality. Use this inequality to give an easy proof that for all f ∈ R

,Recall that in the Riemann Lebesgue lemma

∈ Rso while this exercise is easier, it lacks the generality of the earlier proof. - Suppose f= X
_{[a,b] }. ShowUse this to construct a much simpler proof of the Riemann Lebesgue lemma than that given in the chapter. Hint: Show it works for f a step function and then obtain the conclusion for

∈ R. - Let f= x for x ∈and extend to make the resulting function defined on ℝ and periodic of period 2π. Find the Fourier series of f. Verify the Fourier series converges to the midpoint of the jump and use this series to find a nice formula for. Hint: For the last part consider x =.
- Let f= x
^{2}onand extend to form a 2 π periodic function defined on ℝ. Find the Fourier series of f. Now obtain a famous formula forby letting x = π. - Let f= cos x for x ∈and define f≡−cosx for x ∈. Now extend this function to make it 2 π periodic. Find the Fourier series of f.
- Suppose f,g ∈ R. Show
where α

_{k}are the Fourier coefficients of f and β_{k}are the Fourier coefficients of g. - Recall the partial summation formula, called the Dirichlet formula which says
that
Here A

_{q}≡∑_{k=1}^{q}a_{k}. Also recall Dirichlet’s test which says that if lim_{k→∞}b_{k}= 0, A_{k}are bounded, and ∑converges, then ∑ a_{k}b_{k}converges. Show the partial sums of ∑_{k}sinkx are bounded for each x ∈ ℝ. Using this fact and the Dirichlet test above, obtain some theorems which will state that ∑_{k}a_{k}sinkx converges for all x. - Let be a sequence of positive numbers having the property that lim
_{n→∞}na_{n}= 0 and for all n ∈ ℕ, na_{n}≥a_{n+1}. Show that if this is so, it follows that the series, ∑_{k=1}^{∞}a_{n}sinnx converges uniformly on ℝ. This is a variation of a very interesting problem found in Apostol’s book, [3]. Hint: Use the Dirichlet formula of Problem 10 on ∑ ka_{k}and show the partial sums of ∑are bounded independent of x. To do this, you might argue the maximum value of the partial sums of this series occur when ∑_{k=1}^{n}coskx = 0. Sum this series by considering the real part of the geometric series, ∑_{k=1}^{q}^{k}and then show the partial sums of ∑are Riemann sums for a certain finite integral. - The problem in Apostol’s book mentioned in Problem 11 does not require na
_{n}to be decreasing and is as follows. Let_{k=1}^{∞}be a decreasing sequence of nonnegative numbers which satisfies lim_{n→∞}na_{n}= 0. Thenconverges uniformly on ℝ. You can find this problem worked out completely in Jones [23]. Fill in the details to the following argument or something like it to obtain a proof. First show that for p < q, and x ∈

,(11.26) To do this, use summation by parts using the formula

which you can establish by taking the imaginary part of a geometric series of the form ∑

_{k=1}^{q}^{k}or else the approach used above to find a formula for the Dirichlet kernel. Now defineThus b

→ 0, bis decreasing in p, and if k ≥ n, a_{k}≤ b∕k. Then from 11.26 and the assumptionis decreasing, where this uses the inequalitiesThere are two cases to consider depending on whether x ≤ 1∕q. First suppose that x ≤ 1∕q. Then let m = q and use the bottom line of 11.27 to write that in this case,

If x > 1∕q, then q > 1∕x and you use the top line of 11.27 picking m such that

Then in this case,

_{k}sinkx form a uniformly Cauchy sequence and must converge uniformly on. Now explain why this implies the series converges uniformly on ℝ. - Suppose f= ∑
_{k=1}^{∞}a_{k}sinkx and that the convergence is uniform. Recall something like this holds for power series. Is it reasonable to suppose that f^{′}= ∑_{k=1}^{∞}a_{k}k coskx? Explain. - Suppose ≤ K
_{k}for all x ∈ D whereShow that ∑

_{k=−∞}^{∞}u_{k}converges converges uniformly on D in the sense that for all ε > 0, there exists N such that whenever n > N, - Let a
_{k},b_{k}≥ 0. Show the Cauchy Schwarz inequalityTo do this note that

for all t. Now pick an auspicious value of t, perhaps the value at which p

achieves its minimum. - Suppose f is a differentiable function of period 2π and suppose that both
f and f
^{′}are in Rsuch that for all x ∈and y sufficiently small,Show that the Fourier series of f converges uniformly to f. Hint: First show using the Dini criterion that S

_{n}f→ ffor all x. Next let ∑_{k=−∞}^{∞}a_{k}e^{ikx}be the Fourier series for f. Then from the definition of a_{k}, show that for k≠0,a_{k}=a_{k}^{′}where a_{ k}^{′}is the Fourier coefficient of f^{′}. Now use the Bessel’s inequality to argue that ∑_{k=−∞}^{∞}^{2}< ∞ and then show this implies ∑< ∞. You might want to use the Cauchy Schwarz inequality in Problem 15 to do this part. Then using the version of the Weierstrass M test given in Problem 14 obtain uniform convergence of the Fourier series to f. - Let f be a function defined on ℝ. Then f is even if f= ffor all θ ∈ ℝ. Also f is called odd if for all θ ∈ ℝ, −f= f. Now using the Weierstrass approximation theorem show directly that if h is a continuous even 2π periodic function, then for every ε > 0 there exists an m and constants, a
_{0},,a_{m}such thatfor all θ ∈ ℝ. Hint: Note the function arccos is continuous and maps

onto. Using this show you can define g a continuous function onby g= hfor θ on. Now use the Weierstrass approximation theorem on. - Show that if f is any odd 2π periodic function, then its Fourier series can be simplified
to an expression of the form
and also f

= 0 for all m ∈ ℕ. - Consider the symbol ∑
_{k=1}^{∞}a_{n}. The infinite sum might not converge. Summability methods are systematic ways of assigning a number to such a symbol. The n^{th}Ceasaro mean σ_{n}is defined as the average of the first n partial sums of the series. Thuswhere

Show that if ∑

_{k=1}^{∞}a_{n}converges then lim_{n→∞}σ_{n}also exists and equals the same thing. Next find an example where, although ∑_{k=1}^{∞}a_{n}fails to converge, lim_{n→∞}σ_{n}does exist. This summability method is called Ceasaro summability. Recall the Fejer means were obtained in just this way. - Let 0 < r < 1 and for f a periodic function of period 2π where f ∈ R, consider
where the a

_{k}are the Fourier coefficients of f. Show that if f is continuous, thenHint: You need to find a kernel and write as the integral of the kernel convolved with f. Then consider properties 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 to switch the sum with an integral.

- Recall the Dirichlet kernel is
and it has the property that ∫

_{−π}^{π}D_{n}dt = 1. Show first that this impliesand this implies

Next change the variable to show the integral equals

Now show that

Next show that

Finally show

This is a very important improper integral.

- To work this problem, you should first review Problem 48 on Page 627 about
interchanging the order of iterated integrals. Suppose f is Riemann integrable on every
finite interval, bounded, and
Show that

both exist. Define

Now suppose the Dini condition on f, that

is a function in R

. This happens, for example if for t > 0 and small,