In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that will suffice?
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2$\begingroup$ Why Tchebyshev? (perhaps "Chebyshev" but never mind; why such polynomials are associated with Chebyshev? is my q.). $\endgroup$– Włodzimierz HolsztyńskiCommented May 7, 2015 at 9:25
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2$\begingroup$ It should be Ch...sh with English conventions or Tch...ch with French conventions. Most French people actually mistakenly pronounce "Tchebytchev", by some contagious error [I only refer to the consonants]. Actually Tch...sh is absurd, but is a quite natural error for French-speaking people who have the good convention in mind, because "ch" is very ambiguous in foreign names. $\endgroup$– YCorCommented May 7, 2015 at 21:06
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$\begingroup$ On the spelling question, I am reminded of math.tamu.edu/~boas/courses/math696/spelling-lesson.html. $\endgroup$– KConradCommented May 14, 2015 at 10:05
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$\begingroup$ (sorry: in my previous comment, I mean "the good prononciation in mind") $\endgroup$– YCorCommented May 14, 2015 at 13:22
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