Return to Question

4 added 5 characters in body

For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define $$D(f,g) = \sup_{x \in [-1,1]} \left|{\frac{f(x)-g(x)}{f(x)}}\right|$$ (in this context, let's take $0/0$ to be $0$ and $x/0 = \infty$ for every $x \not= 0$). I am looking for a theory of approximation by polynomials with respect to this error that parallels the existing theory of approximation by polynomials with respect to the $L_\infty$-norm.

For example: is it true that for every real $f$ continuous on $[-1,1]$ and every degree $n$ there exists a best approximating polynomial $p$ of degree $n$ in the sense that $D(f,p)$ is minimal among all polynomials of degree $n$? In such a case, can we find it or characterize it?

I would like to know if this has been studied before and where.

Edit: From the answers I got below, perhaps I should add that, in the application I have in mind, $f(x)$ may or may not be $0$ exactly at one place: at $x = 0$. This rules our infinitely many zeros in $[-1,1]$, and in case $f(0) = 0$, the best approximator $p(x)$ should be such that $p(0) = 0$ (my convention to take $0/0 = 0$ and $x/0 = \infty$ for $x \not= 0$ comes from this).

3 added 357 characters in body; added 29 characters in body

For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define $$D(f,g) = \sup_{x \in [-1,1]} \left|{\frac{f(x)-g(x)}{f(x)}}\right|$$ (in this context, let's take $0/0$ to be $0$ and $x/0 = \infty$ for every $x \not= 0$). I am looking for a theory of approximation by polynomials with respect to this error that parallels the existing theory of approximation by polynomials with respect to the $L_\infty$-norm.

For example: is it true that for every real $f$ continuous on $[-1,1]$ and every degree $n$ there exists a best approximating polynomial $p$ of degree $n$ in the sense that $D(f,p)$ is minimal among all polynomials of degree $n$? In such a case, can we find it or characterize it?

I would like to know if this has been studied before and where.

Edit: From the answers I got below, perhaps I should add that, in the application I have in mind, $f(x)$ may or may not be $0$ exactly at one place: at $x = 0$. This rules our infinitely zeros in $[-1,1]$, and in case $f(0) = 0$, the best approximator $p(x)$ should be such that $p(0) = 0$ (my convention to take $0/0 = 0$ and $x/0 = \infty$ for $x \not= 0$ comes from this).

2 Moved the absolute bars outside the fraction.

For given continuous real functions $f$ and $g$ defined on $[-1,1]$, let's define $$D(f,g) = \sup_{x \in [-1,1]} \frac{| f(x)-g(x) |}{|f(x)|} left|{\frac{f(x)-g(x)}{f(x)}}\right|$$ (in this context, let's take $0/0$ to be $0$ and $x/0 = \infty$ for every $x \not= 0$). I am looking for a theory of approximation by polynomials with respect to this error that parallels the existing theory of approximation by polynomials with respect to the $L_\infty$-norm.

For example: is it true that for every real $f$ continuous on $[-1,1]$ and every degree $n$ there exists a best approximating polynomial $p$ of degree $n$ in the sense that $D(f,p)$ is minimal among all polynomials of degree $n$? In such a case, can we find it or characterize it?

I would like to know if this has been studied before and where.

1