Intereting Posts

Matrices with real entries such that $(I -(AB-BA))^{n}=0$
Ordered tuples of proper classes
Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$
Aggregate arrivals from a renewal process
Evaluation of $\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$
Prove quotient is a perfect square
What are the consequences if Axiom of Infinity is negated?
Find the density of the sum of two uniform random variables
Show that $\frac{G \oplus G \oplus G }{ (1,0,-1)G + (0,1,-1)G} = G$
On a decreasing sequence of compact subsets of a Hausdorff topological space
Diophantine equation involving factorial …
“Real” cardinality, say, $\aleph_\pi$?
If $ H $ is a normal subgroup of $ G $, is $ G/H \times H \cong G $?
how to parameterize a curve f(x,y)?
$\frac{dx}{dt}=-\lambda x +\epsilon x(t-a)$ series solution via Laplace method

I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if $f:[0,\pi]\to \mathbb{C}$ is $C^1$ and $f(0)=f(\pi)=0$ then

\begin{equation}

\int_0^\pi |f(t)|^2 dt \leq \int_0^\pi |f'(t)|^2 dt.

\end{equation}

See link.

The proof that I know starts by proving that if

$$ \int_0^{2\pi} F(t) dt =0 $$

then

$$

\int_0^{2\pi} |F(t)|^2 dt \leq \int_0^{2\pi} |F'(t)|^2 dt.

$$

using Parseval’s identity. From this, one proves the desired inequality for $f$ on $[0,\pi]$ by “extending” $f$ to an odd $C^1$ function on $[-\pi,\pi]$.

Are there other proofs? (Straightforward or otherwise…)

- Are there other kinds of bump functions than $e^\frac1{x^2-1}$?
- Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?
- What is the single most influential book every mathematician should read?
- Good books on Philosophy of Mathematics
- How many different definitions of $e$ are there?
- Elementary Papers at ArXiv

- How does linear algebra help with computer science
- Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities
- Who are some blind or otherwise disabled mathematicians who have made important contributions to mathematics?
- Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
- Best intuitive metaphors for math concepts (of any level)
- Examples of open problems solved through short proof
- Fourier transform and non-standard calculus
- Nice examples of groups which are not obviously groups
- An inequality by Hardy
- Fourier transform is uniformly continuous

If you are willing to get a **non-sharp constant**, here’s another proof found in many differential geometry texts. Without loss of generality assume $f \geq 0$. (Replacing $f$ by $|f|$ doesn’t change the integrals on either side, if $f$ is assumed to be $C^1$.)

Let $2M = \sup f$, and let $t_0 \in (0,\pi)$ attain this maximum.

Let $X(t) = f(t) – M$ and $Y(t) = \sqrt{M^2 – X(t)^2}$ if $t \leq t_0$ and $-\sqrt{M^2 – X(t)^2}$ if $t \geq t_0$.

We have that $(X(t),Y(t))$ lies on the circle of radius $M$, and goes around the circle exactly once as $t$ goes from $0$ to $\pi$. We thus can use a well-known formula to conclude that

$$ -\int_0^\pi Y(t) X'(t) \mathrm{d}t = \text{Area of disk} = \pi M^2 $$

By Schwarz inequality, however, we have

$$ \int_0^\pi Y(t) X'(t) \mathrm{d}t \leq \sqrt{ \int_0^\pi Y^2\mathrm{d}t \int_0^\pi X’^2\mathrm{d}t} = \sqrt{ \left(\pi M^2 – \int_0^\pi X^2\mathrm{d}t \right) \int_0^\pi X'(t)^2\mathrm{d}t }$$

Squaring we get

$$ \pi^2 M^4 \leq \left(\pi M^2 – \int_0^\pi X^2 \mathrm{d}t\right) \int_0^\pi f’^2\mathrm{d}t $$

Now, notice that

$$ \int_0^\pi f^2 ~\mathrm{d}t = \int_0^\pi (X + M)^2 ~\mathrm{d}t = \pi M^2 + \int_0^\pi X^2 ~\mathrm{d}t + 2M \int_0^{\pi} X ~\mathrm{d}t \leq \pi M^2 (1+A)^2 $$

where

$$ A: = \left[ \frac{1}{\pi M^2} \int_0^\pi X^2 ~\mathrm{d}t \right] < 1. $$

This implies

$$ \int_0^\pi f^2 ~\mathrm{d}t \leq (1 + A)^2(1-A^2) \int_0^\pi |f’|^2 ~\mathrm{d}t$$

The coefficient has a maximum when $A = 1/2$ or that

$$ \int_0^\pi f^2 ~\mathrm{d}t \leq \frac{27}{16} \int_0^\pi |f’|^2~\mathrm{d}t $$

If $\int_0^\pi X ~\mathrm{d}t = 0$, we can sharpen the coefficient to $(1 + A^2)(1-A^2) = 1 – A^4 \leq 1$. This can be achieved by extending $f$ to a function $g$ on $(-\pi,\pi)$ with an odd extension, exactly as you have described for the Fourier proof.

The following proof is found in section 7.7 of Hardy-Littlewood-Polya *Inequalities*, it is motivated by Hilbert’s investigations into calculus of variations, especially Hilbert’s method of invariant integrals.

Consider the expression

$$ (y’^2 – y^2) – (y’ – y\cot x)^2 = -(1+ \cot^2 x) y^2 + 2y y’ \cot x $$

So

$$ \left[(y’^2 – y^2) – (y’ – y\cot x)^2 \right]\mathrm{d}x = -(\csc^2 x)y^2 \mathrm{d}x + 2y \mathrm{d}y \cot x = \mathrm{d} ( y^2 \cot x )$$

Now, since $y’ \in L^2$, we have that

$$ y^2(x) = \left(\int_0^x y'(s) \mathrm{d}s\right)^2 \leq \int_0^x y'(s)^2 \mathrm{d}s \int_0^x 1\mathrm{d}s \leq x \int_0^x y'(s)^2 \mathrm{d}s $$

So we have that

$$ \frac{y^2(x)}{x} = o(1) $$

and hence $y = o(\sqrt{x})$. Similarly we have that $y^2$ approaches 0 superlinearly at $\pi$. This implies that $\lim_{x\to \{0,\pi\}} y^2 \cot x = 0$. Hence the exact integral

$$ \int_0^\pi (y’^2 – y^2) – (y’ – y\cot x)^2 \mathrm{d}x = \int \mathrm{d}\left( y^2 \cot x\right) = 0 – 0 = 0 $$

Therefore we have

$$ \int_0^\pi (y’^2 – y^2) \mathrm{d}x = \int_0^\pi (y’ – y\cot x)^2 \mathrm{d}x \geq 0 $$

with equality only if

$$ y’ = y \cot x $$

which is when $y = k \sin x$.

You can find different proof in the book of B. Dacorogna ‘Introduction to the calculus of variations’.

- Is a prime ideal in the polynomial ring over an algebraically closed field prime also in polynomial rings over extension fields?
- Always a differentiable path through a convergent sequence of points in $\mathbb{R}^n$?
- Complete statistic: Poisson Distribution
- Self study text for Elementary Number theory
- Question about Feller's book on the Central Limit Theorem
- Flip a coin until a head comes up. Why is “infinitely many tails” an event we need to consider?
- Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
- $\infty = -1 $ paradox
- Find this limit without using L'Hospital's rule
- Subgroups whose order is relatively prime to the index of another subgroup
- Intuitive explanation for integration
- Fibered products in $\mathsf {Set}$
- Converse of Lagrange's theorem for abelian groups
- $e$ as the limit of a sequence
- Automorphism group of ${\bf Z}_p$