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Hello,

I am very new to the field of approximation theory, and since an extended search on the Internet did not provide answers for two rather basic questions, I decided to ask them here.

1) From my understanding upper bounds for

$$\inf_{p} inf_{q} \int_{-1}^{1} |f(x) - p(x)|^{2pq(x)|^{2p} dt$$

with $f$ continuous and $p$ q$a polynomial , of degree$n$, are expressed in terms of the$L^p$smoothness of$f$and the degree in terms of the polynomial degree$p$. n$. Could somebody point me to a proof of such a result?

2) Heuristically, what kind of information do lower bounds for the above infinum contain ? (For example, suppose that I can give a lower bound of $p!$ for the above infinimum as $p \rightarrow \infty$).

My last question might not be well-posed, so if it doesn't make sense please ignore it.

Thank you.

1

# Polynomial approximation in L^p norms

Hello,

I am very new to the field of approximation theory, and since an extended search on the Internet did not provide answers for two rather basic questions, I decided to ask them here.

1) From my understanding upper bounds for

$$\inf_{p} \int_{-1}^{1} |f(x) - p(x)|^{2p} dt$$

with $f$ continuous and $p$ a polynomial, are expressed in terms of the $L^p$ smoothness of $f$ and the degree of the polynomial $p$. Could somebody point me to a proof of such a result?

2) Heuristically, what kind of information do lower bounds for the above infinum contain ? (For example, suppose that I can give a lower bound of $p!$ for the above infinimum as $p \rightarrow \infty$).

My last question might not be well-posed, so if it doesn't make sense please ignore it.

Thank you.