Let denote the usual inner product of . Evaluate the integral
where is a positive symmetric matrix and .
is a positive symmetric matrix , so is
. For a positive symmetric matrix
there exists an
positive symmetric matrix such that
. Applying this to
our integral becomes
where is the Euclidean norm. Applying a change of variables we have that
Since then by converting to polar coordinates we have that
Here denotes the surface area measure of the unit sphere and it is known to be
where denotes the Gamma Euler function for which it holds that
Let be a measurable of finite length set. Evaluate the limit:
Let us begin with the Fourier series of
which is of the form:
Integrating ( 1 ) we get that
since it follows from Riemann – Lebesgue lemma that
Let denote the -th harmonic number. Prove that forall it holds that
Well we are stating two lemmata.
Lemma 1: For all it holds that
Proof: Pretty straight forward calculations show that
and Lemma 1 is proved.
Lemma 2: For all it holds that
Proof: We begin by lemma 1 and successively we have
and Lemma 2 follows.
Now, mapping back at Lemma we have that
Integrating we have that
Let be a continuous function such that
where . Prove that is constant.
Let and consider the function
where and . If for all then prove that
is differentiable in
and the derivative is given by
Hence we want to prove that . We note that forall it holds that
and the conclusion follows.