# Exercises for march 26 and march 27th entries

Here are the exercises for the last two posts, Convolution and good kernels and Cesàro summability and Fejér’s theorem.

Exercise 1:Let be a real-valued function which is -periodic and integrable on any finite interval. Show that, forAnd that

**Solution:**

Let be defined as Then and by the periodicity of we have So

where is a primitive for Indeed, by the chain rule we have so we only need to apply the fundamental theorem of calculus. But again by the fundamental theorem of calculus, we have

so we may conclude.The same argument gives the second equality of the first point. What this means is that for such a function the integral of over an integral is invariant by a translation of length equal to the period of

For the second point, note that with a similar argument as above, we get

for As for the second equality, we have

However, applying what we learned in the first set of equalities of this exercise, we can translate the domain of integration of the first term in the right side by and get

hence

QED

Exercise 2:Show that if a series of complex numbers converges to , then it is Cesàro summable to

**Solution:**

Let be a series converging to Denoting the partial sums of this series by we need to show that as

Since for every there exists such that for all So for large enough we get

where is an upper bound for Since and as we have

for any hence as wanted.

QED

Exercise 3:Show that Lagrange’s trigonometric identities hold. i.e.and

**Solution:** I’ll actually only post the solution to the first identity. The idea is that is the real part of which can be written as

We then multiply both the numerator and denominator by the complex conjugate of the denominator. The denominator will then be real and given by:

while the real part of the numerator will be given by

Now, note that the expression in brackets is equal to

Putting the numerator over the denominator, we thus get

QED

I actually use the other identity too in the next exercise but I am too lazy to write it down.

Exercise 4:Show that the Fejér kernel is given by

Recall that the Fejér kernel is by definition where is the n-th Dirichlet kernel. Well, in Exercise 5, I’ll show that

So we get

where the last equality is justified by the identities of Exercise 3. This is equal to

Which gives exactly what we wanted.

QED

Exercise 5:Show that the N-th Dirichlet kernel is given by

**Solution:**

Recall that by definition Let Then since

and

we get

Multiplying by we thus get

But since this last equality gives

as we wanted.

QED

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