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Tagged with chebyshev-polynomials functional-analysis
10
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Approximating Chebyshev Polynomials by Truncated Fourier Series
It is known (see e.g. https://www.math.ucdavis.edu/~bremer/classes/fall2018/MAT128a/lecture9.pdf) that any continuous function $f: [-1,1] \to \mathbb{C}$ admits a Chebyshev expansion
$$
f(x) = \sum_{n=...
- 191
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Chebyshev Polynomial of degree 3
This is the question:
(a) Given a function $f(x)=e^{-(a+1)x}$. Find the polynomial of degree $2$ , $p_{2}(x)$ such that (with $w(x)=1$):
$$||f-p_{2}||_{L^{2}(0,1)}=\min_{q \in P_{2}(0,1)}||f-q||_{L^{2}...
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Show that the lowest-norm monic polynomial is of the form $\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\right)$
Let $T_n\in P_n[-1,1]$ the n-th Chebyshev polynomial. Show that the lowest-norm monic polynomial in $P_n[a,b]$ is of the form $$\frac{(b-a)^n}{2^n}\frac{1}{2^n}T_n\left(\frac{2}{b-a}x-\frac{b+a}{b-a}\...
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Spectrum self-adjoint operator induced by right shift operator on $\ell^2(\mathbb{N})$.
Let $e_i$ be the standard bais vectors of $\ell^2(\mathbb{N)}$ and let $S$ denote the right shift operator on $\ell^2(\mathbb{N)}$, i.e. $Se_i= e_{i+1}$. Now the operator $T = S + S^*$ is self-adjoint ...
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Showing That Chebyshev Polynomials Are Orthogonal
This is a problem in an upcoming lecture:
Show that the first two Chebyshev polynomials, $T_0(x) = 1$ and $T_1(x) = x$ are orthogonal with respect to the weighting function $r(x) = (1 − x^2)^{-\...
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some problem about chebyshev series
Suppose that $f \in C[-1,1]$ has a chebyshev series $\sum_{n=1}^{\infty}a_nT_n$
(b) show that $E_n(T_{n+1})=1$
(c) show that $|E_n(f)-|a_{n+1}|| \le \sum_{k=n+2}^{\infty}|a_k|$
cf : $E_n(f)= \inf\{||...
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Proving a bound for the leading coefficient of a polynomial.
Show that every real polynomial $x\in C[a,b]$ of degree $n\ge 1$ with leading term $\beta_n t^n$ satisfies $$||x||\ge |\beta_n|\frac{(b-a)^n}{2^{2n-1}}.$$
I am having difficulty proving this. Here on ...
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Prove that the Chebyshev polynomials have interlacing zeros.
Between any two neighboring zeros of $T_n$ there is precisely one zero of $T_{n-1}$. Prove this property. Where $$T_n(t) = \cos(n\theta), \,\,\,\theta = \arccos(t),\,\,\,\,n=0,1,2,\dots$$
I have the ...
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2
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2
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Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum
The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation
$$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$
Alternatively they are defined inductively:
$$U_0(x)=1 \...
- 22k
1
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1
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Link between Chebyshev polynomials and best approximants
I'm reading Interpolation and Approximation by Davis, more specifically "Best Approximation" Chapter VII.
Let $n \in \mathbb N$.
Let $C[a,b]$ denote the set of continuous real functions ...
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