I have tried to solve the integral $${\int_0^1 U_n (x) U_m (x)x dx },$$ where ${U_n (x) }$ denotes Chebyshev polynomial of the second kind. Solving the integration and checking the result, I noticed that it does not match the ones I did analytically for $n=0$, $m=1$; and some other values, so, there must be […]

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post suggests, I’m asking the general problem: given an $n$-th degree polynomial $p(x)$ on $[-1,1]$, find the best approximation $q(x)$ in lower degree($j<n$) polynomial space: $\min_{\operatorname{deg}q(x)=j} […]

The following $n\times n$ determinant identity appears as eq. 19 on Mathworld’s entry for the Chebyshev polynomials of the second kind: $$U_n(x)=\det{A_n(x)}\equiv \begin{vmatrix}2 x& 1 & 0 &\cdots &0\\ 1 & 2x &1 &\cdots &0 \\ 0 & 1 & 2x &\cdots &0\\0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots […]

In the book Proofs from The Book by Aigner and Ziegler there is a proof of ‘Chebyshev’s Theorem’ which states that if $p(x)$ is a real polynomial of degree n with leading coefficient $1$ then $$ \max_{-1 \leq x \leq 1} |p(x)| \geq \frac{1}{2^{n-1}}$$ In the proof $g( \theta ) = p( \cos \theta )$ […]

$P(x)$ is a monic polynominal ( the highest coefficient is 1 ). $deg P(x) = n$. Prove that exist a constant $c$ that $P(x)+c.T_n(x)$ has all its roots real. $T_{n-1}(x)$ is the $n$th Chebyshev polynomial Please help me :((

Express First kind Chebyshev polynomial in terms of monomials First kind Chebyshev polynomial of order n ($T_n$) is defined in terms of cosine function as follow: 1) $T_n(\cos x)=\cos n x$ Instead of working with cosine functions, I want to express Chebyshev polynomial in terms of monomials (powers of x) using the following recursion formula: […]

I have competing answers on my question about “Returning Paths on Cubic Graphs Without Backtracking”. Assuming Chris is right the following should work. Up to one thing: The number of returning paths on 3-regular graphs of length $r$ without backtracking may be written as $2^{-r/2}p_r(x/\sqrt{2})$ which is a Chebyshev Polynomial of the Second Kind $U_r(x)$. […]

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