I would like to know if this has been studied before and where. Then $$\frac{|f(x)-q(x)p(x)|}{|f(x)|} =|1-\frac{p(x)}{h(x)}|.$$ If $\phi$ is the best approximation to $1$ in $V$, then $q\cdot \phi$ will be a best approximation to $f$ of degree $n+\hbox{degree}(q)$ in the relative error sense. But it is more practical to minimize an L2 norm using orthogonal polynomials. Edit in response to your comment If $f$ has a finite number of zeroes in $[-1,1]$ and can be written as $f=q\cdot h$ where $q$ is a polynomial and $h$ is

I can accept your answer now. –slimton Oct 10 '10 at 20:24 add a comment| up vote 2 down vote Take f to be non-zero but have infinitely many zeros in Generated Sat, 15 Oct 2016 19:37:12 GMT by s_wx1131 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.5/ Connection Browse hundreds of Advanced Math tutors. Of course also various existence claims on these are needed share|cite|improve this answer answered Oct 7 '10 at 17:12 Helge 2,6081420 I do not think your example is correct.

Moreover, each $\phi\in V$ has at most $n$ zeros in $[-1,1]$, so that it satisfies what is called the Haar condition. 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. Generated Sat, 15 Oct 2016 19:37:12 GMT by s_wx1131 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.7/ Connection 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

I'll have to reread JuliÃ¡n's comments on his excellent answer. The system returned: (22) Invalid argument The remote host or network may be down. Then you ask for the existence of $P\in\mathbb{P}(n)$ such that $$\|1-P/f\|=\inf_{p\in\mathbb{P}(n)}\|1-p/f\|.$$ This can be brought under the theory of approximation in the $L^\infty$ norm. Please try the request again.

The original problem is now recast as: find the best approximation in $V$ of the constant function $1$. On the other hand if f is bounded from below away from zero there will at least exist a sequence of polynomials such that the relative error converges to zero, by All rights reserved. Sometimes there is a "best of the best" example which is the limit of the best $L_p$ approximations as p goes to $\infty$.

Each $\phi_k$ is a continuous function, they are independent and generate an $n$-dimensional subspace of the space of continuous functions on $[-1,1]$ that we denote by $V$, which is nothing but It might be that for uniqueness you need to discriminate how much time (or the number of times) the sup is obtained. So if you want to allow a zero, then you need to have a condition of the form that the first $k$ derivatives are $0$ and the $k+1$st is non-zero. share|cite|improve this answer answered Oct 6 '10 at 15:32 Steven Heston 396511 add a comment| up vote 0 down vote I don't see Helge's example as a problem.

This means that $|\frac{f-g}{f}|$ has limit 1 as we approach 0 (since g has bounded derivative at 0) so 1 is the best we could hope for. Your cache administrator is webmaster. Find bounds for the error and relative error in approximating sin ( sqrt(2)) by sin ( 1.414) Expert Answer Get this answer with Chegg Study View this answer OR Find your In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms

share|cite|improve this answer answered Oct 7 '10 at 18:53 Aaron Meyerowitz 17.9k12366 add a comment| Your Answer draft saved draft discarded Sign up or log in Sign up using Google Generated Sat, 15 Oct 2016 19:37:12 GMT by s_wx1131 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection I believe that when one chooses polynomials to approximate functions in practice (in calculators for example) one uses the relative error as the objective function. In such a case, can we find it or characterize it?

Please try the request again. Is the $D(\cdot, \cdot)$ you've defined even a metric? –Jerry Gagelman Oct 6 '10 at 15:39 I don't know if it is a metric: I just look for a Not the answer you're looking for? polynomials approximation-theory share|cite|improve this question edited Oct 7 '10 at 11:14 asked Oct 6 '10 at 11:30 slimton 19319 Whether you're considering linear approximation (like Chebyshev approximation for continuous

In the following I assume that $f(x)\ne0$ for all $x\in[-1,1]$. The system returned: (22) Invalid argument The remote host or network may be down. Over 6 million trees planted current community chat MathOverflow MathOverflow Meta your communities Sign up or log in to customize your list. Please try the request again.

I wonder if something can be said in that case. Generated Sat, 15 Oct 2016 19:37:12 GMT by s_wx1131 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection A polynomial can only have a finite number of zeros so clearly the D(f,p) you defined will be infinite for every polynomial. Question: Find bounds for the error and relative error in ap...

The Legendre polynomials will minimize the average squared approximation error on [-1,1]. Browse other questions tagged polynomials approximation-theory or ask your own question. Theoretically you can approximate continuous functions arbitrarily closely by polynomials. 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

Finally, Remes' algorithm can be used to construct the best approximation. Generated Sat, 15 Oct 2016 19:37:12 GMT by s_wx1131 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection share|cite|improve this answer edited Oct 6 '10 at 16:56 answered Oct 6 '10 at 14:23 Johan Richter 213 add a comment| up vote 1 down vote Consider $$ f(x) = \sqrt{|x|}. However, in the application I have in mind, the function $f(x)$ may or may not be $0$ exactly at one place: at $x = 0$.

asked 6 years ago viewed 1459 times active 6 years ago Visit Chat Get the weekly newsletter! Your cache administrator is webmaster. Beyond these special cases I do not know anything but presumably relative approximation will have been studied before. The system returned: (22) Invalid argument The remote host or network may be down.

Your cache administrator is webmaster. The general theory of approximation shows that there is a unique best approximation, and that it can be characterized as follows: $\phi\in V$ is the best approximation in $V$ of the Related 6Polynomial upper approximation with respect to the Gaussian measure10The closures in $C^0(\mathbb{R}, \mathbb{R})$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients7Uniform approximation of $x^n$ by Perhaps I should look at the generators $(x-x_0)^k/f(x)$ instead. –slimton Oct 7 '10 at 7:59 The trick to deal with finitely many zeros is wonderful.