## Level sets for Chebyshev polynomials

Click on the links below to see some pictures of the level sets

$\{ (x,y) \in [-1,1]^2: n|T_n(x)-T_n(y)| > \varepsilon\}$

for some values of the parameters $n \geq 1, \varepsilon >0.$
Figure 1
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Figure 5

### 4 Responses to “Level sets for Chebyshev polynomials”

1. Miguel Lacruz Says:

Here is an attempt to apply dilation theory in conjecture 9. The idea is that any contraction $T$ on the Hilbert space $\ell^2({\rm {\bf N}})$ has a unitary power dilation, that is, $\ell^2({\rm {\bf N}})$ is a subspace of a larger Hilbert space $H$ and there is a unitary operator $U:H \rightarrow H$ with the nice property that $PU^nP=T^n$ for all $n \geq 0$, where $P:H \rightarrow \ell^2({\rm {\bf N}})$ represents the orthogonal projection. This suggests the following modification of conjecture 9:

Claim A There exists $N>0$ and there are unit vectors $f,g \in H$ such that

$\displaystyle \max \{ \|(I-P_N)f \| , \|(I-P_N)g\| \} < \varepsilon$,

$\displaystyle |\langle U^nf,g \rangle|<\frac{1}{F(N)}$ for each $0 \leq n \leq F(N)$,

where $P_N:H \rightarrow \ell^2(N)$ denotes the orthogonal projection.

Now, consider the vectors $v=Pf$, $w=Pg,$ and notice that

$\displaystyle \langle U^nPf,Pg \rangle = \langle U^nf,g \rangle + \langle U^n(Pf-f),g \rangle + \langle U^nPf,Pg-g \rangle$

and therefore

$\displaystyle |\langle T^nv,w \rangle| = | \langle U^nPf,Pg \rangle | < \frac{1}{F(N)} + 2 \varepsilon,$

for all $0 \leq n \leq F(N).$ I wonder if this kind of argument can be turned into a proof that FISP2 is a consequence of claim A. Notice that the spectral theorem can be applied to represent the unitary operator $U$ on the Hilbert space $H= L^2(\mu)$ as a multiplication operator, that is, $Uf= \varphi \cdot f$ where $\varphi$ is a measurable function and $|\varphi (z)|=1$ a.e.

It is clear that claim A has the following equivalent formulation:

Claim B There exists $N>0$ and there is a function $h \in L^2(\mu)$ such that

$\displaystyle \|(I-P_N)h \| < \varepsilon$,

$\displaystyle \left |\int \varphi^n h^2 d\mu \right |<\frac{1}{F(N)}$ for each $0 \leq n \leq F(N).$

2. Miguel Lacruz Says:

Claim. If $U$ is a unitary operator on $H$ then for every $\varepsilon >0$ there are unit vectors $f,g \in H$ such that

$\displaymath{\|P_Nf \| \geq 1 - \varepsilon^2$, $\|P_Nfg\| \geq 1 - \varepsilon^2}$

3. Miguel Lacruz Says:

Claim. If $U$ is a unitary operator on $H$ then for every $\varepsilon >0$ there are unit vectors $f,g \in H$ such that

$\|P_Nf \| \geq 1 - \varepsilon^2, \|P_Nfg\| \geq 1 - \varepsilon^2}$

4. Miguel Lacruz Says:

Claim. If $U$ is a unitary operator on $H$ then for every $\varepsilon >0$ there are unit vectors $f,g \in H$ such that

$\|P_Nf \| \geq 1 - \varepsilon^2, \|P_Ng\| \geq 1 - \varepsilon^2}$