Talk:Approximation theory

Splines

"Modern mathematical libraries often reduce the domain into many tiny segments and use a low-degree polynomial for each segment." Are those technically Splines? If so it would be good to link to them.38.98.147.133 (talk) 15:13, 4 December 2012 (UTC)[reply]

Definition and scope

Given the definition:

"In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby."

... wouldn't Power series, particularly Taylor series fit within this realm? And how about Newtons method? Vonkje 07:22, 5 October 2005 (UTC)[reply]

I think that approximation theory is a huge field and that any page with such a title can only outline the main areas. The article at present goes into too much detail in certain topics. For example, there should be a separate page on Chebyshev approximation and maybe even on the Remes Algorithm. (There is already such a page but it is not very good yet.) Any comments? Should I make these changes? --Pftupper 02:33, 4 November 2006 (UTC)[reply]

I agree that the treatment is uneven and Chebyshev and Remes deserve their own pages. To my mind, power series are more for representation than approximation, but it depends on just how one uses the term. Certainly, lots of numerical methods are derived by starting with a Taylor series approximation. As to Newton's method, I'd say not her. My opinion is, make the changes! JJL 03:19, 5 November 2006 (UTC)[reply]

As the original author of all the Chebyshev and Remes stuff on this page, I agree fully with the suggestion to reorganize the subject. I think that what I put in about Chebyshev and Remes is really cool stuff, and I didn't see it anywhere, so I put it in. I knew that "approximation theory" isn't really the right category, but there was essentially no material on the page of that name, and other pages were linking to it, so I decided to put it there.

A few other bits of information: The "computer" category goes into incredible detail in all the many many subtopics. Perhaps this material should be renamed "computer approximations" and considered to be part of that category tree. And then "approximation theory" (this page) should be devoted to the more pure maths aspects (Taylor series, etc.) with a link to "computer approximations" at the opportune moment. Also, there is a page called "function approximation", which appears to be little more than a stub. Its future should be taken into account also. William Ackerman 16:47, 6 November 2006 (UTC)[reply]

Q: would a link to Karhunen–Loève transform / Principal Component analysis belong in this article? I'm no expert but it seems the goal is the same (efficient approximation of a function through a minimum set of parameters); —Preceding unsigned comment added by 66.158.152.170 (talk) 06:33, 18 June 2009 (UTC)[reply]

I think the article should more or less cover the most important topics that are within the scope of the major journals in this area (although, as pointed out above, obviously not in great detail). As far as I know, the Karhunen–Loève theorem and PCA are outside their scope. As it is, the present article deals too exclusively with polynomial functions in one variable – although perhaps also the use of Bernstein polynomials should be mentioned). A paragraph or so on Padé approximants is definitely in order, and some other must topics, however briefly dwelt on, are splines, Fourier analysis, and wavelets. A few words on the approximation of multivariate functions (for example, by Bézier surfaces) are needed to round out the treatment. --Lambiam 16:27, 27 April 2010 (UTC)[reply]

Merge with Approximation theory/Proofs

The article name 'Approximation theory/Proofs' does not conform to Wikipedia naming conventions and the subject does not meet notability guidelines in its own right. That article has other issues that need to be addressed as well. However, the proof does seem to be instructive and significant in that it motivates Remez' algorithm, so I believe that it should be merged with this article. See Wikipedia:WikiProject Mathematics/Proofs for a general discussion of proofs in Wikipedia.--RDBury (talk) 20:49, 1 January 2010 (UTC)[reply]

External links modified

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motivation required

I studied approximation theory, decades ago, when programmable calculators were new and compute time expensive. Today computation power is very cheap and available, and as I address a new problem, it seems that the well-know-problems of poles and "squiggles" (errors) of polynomial approximation renders these as anachronistic toys that should be abandoned. Certainly we don't need to reduce a complex physical chemistry curves into an 6th degree polynomial so that someone can enter this, tediously, on a calculator. I believe the other commenter is correct; splines have value when we MUST interpolate from from empirical data, but poly'-curves are anachorisms. ^^ — Preceding unsigned comment added by 24.165.181.87 (talk) 21:17, 7 July 2019 (UTC)[reply]