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
Figure 2
Figure 3
Figure 4
Figure 5

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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}

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