Talk:Basis function

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[edit] definition

This page should give a definition of a basis function first, bolding the key word. Fresheneesz 05:45, 23 March 2006 (UTC)

[edit] Square integrable

There is a problem about sine and cosine not being square integrable but being used as basis vectors for a space of vectors that are square integrable. If I tried to fix it, the language might not be right. David R. Ingham 01:49, 9 September 2006 (UTC)

[edit] a comment

this page is sloppily written and redundant. it omits essential mathematical details and claims to be mathematics. the topic it purports to discuss is covered in much better fashion in Hamel basis, Hilbert space, and probably other pages. one might wanna consider replacing the math categories by physics ones. (IMHO, it certainly doesn't belong in the functional analysis category) Mct mht 05:53, 9 September 2006 (UTC)

i've removed the article from the functional analysis category. Mct mht 20:45, 23 September 2006 (UTC)

[edit] Basis of a vector space

The point of a basis is that it is a minimal (non-redundant, i.e. linear independent) set of vectors or functions that span a certain space.

E.g. for vectors could be defined as follows:

A system of vectors v_1 , \ldots ,v_r \in V is called Basis of V if they are a spanning set of V and they are linearly independent.

1. v_1 , \ldots ,v_r \in V is a spanning set of a vector space V if V = span(v_1,...,v_r) = \left\{ {\lambda _1 ,v_1 + \cdots + \lambda _r v_r |\lambda _1 , \ldots ,\lambda _r \in \mathbb K} \right\}

2. Linear independence: v_1 , \ldots v_r are linearly independent if \lambda _1 v_1 + \cdots \lambda _r v_r = 0 \Rightarrow \lambda _1 = \cdots \lambda _r = 0.

These statements are all equivalent: - The vectors v_1 , \ldots v_r are a basis.

- Every vector in V is defined by a unique linear combination of the basis vectors.

- If you take out one vector from the basis v_1 , \ldots v_r, it is no longer a spanning set of V.

... And of course the number of basis vectors coinsides with the dimension of the vector space.

—The preceding unsigned comment was added by 195.176.0.51 (talk • contribs) 01:25, 16 September 2006 (UTC)

one does need to be a little careful in the infinite dimensional case with regards to what linear combinations are allowed. Lunch 15:16, 25 September 2006 (UTC)

[edit] In which subject areas is the term basis function used?

(copied from User talk:Mct mht)
i think most mathematicians will agree that the presentation and content is not mathematical. there are a few entries in the edit history that are from mathematicians, with edit summaries like "more work needs to be done". the discussion is in the same vein as what can be found in some physics texts, see for example Intro to Quantum Mechanics (i believe that's the title) by David Griffiths. it's certainly misleading to call it functional analysis. as i said on the talk page, the stuff the article seems to purport to cover is discussed in Hamel basis and Hilbert space in much better fashion. Mct mht 03:52, 25 September 2006 (UTC)

Thanks for the reference. Do you know if Griffiths actually uses the term basis function?
For the record:
Introduction to Quantum Mechanics (2nd Edition) (Hardcover)
by David J. Griffiths (Author)

[edit] References not using the term basis function

I do not find the term basis function in any of these math references:
  • Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional Analysis: An Introduction, American Mathematical Society, 2004.
  • Christopher E. Heil, A basis theory primer, 1997.
  • Glenn James and Robert C. James, Mathematics Dictionary, Fourth Edition, Van Nostrand Reinhold, 1976.
  • Erwin Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1989.
  • Anthony N. Michel and Charles J. Herget, Applied Algebra and Functional Analysis, Dover, 1993.
  • Georgi E. Shilov, Elementary Functional Analysis, Dover, 1996.
  • http://planetmath.org/

[edit] References using the term basis function

However, I find numerous examples of the term basis function or basis functions (and one example of function basis set) being used at:
  • [1] Francis J. Narcowich, Joseph D. Ward and Holger Wendland. Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comp. 74 (2005) 743-763.
  • [2] Christine Bernardi, Tariq Aslam, Jeremy Levesley, H. Niederreiter and Igor Shparlinski. Book Review. Math. Comp. 73 (2004) 1577-1582. (the reviewed book is: Radial basis functions theory and implementations, by M. D. Buhmann )
  • [3] Robert Schaback and Holger Wendland. Inverse and saturation theorems for radial basis function interpolation. Math. Comp. 71 (2002) 669-681.
  • [4] M. D. Buhmann. A new class of radial basis functions with compact support. Math. Comp. 70 (2001) 307-318.
  • [5] Holger Wendland. Meshless Galerkin methods using radial basis functions. Math. Comp. 68 (1999) 1521-1531.
  • [6] R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions . Math. Comp. 68 (1999) 201-216.
  • Haar Function -- from Wolfram MathWorld: "... where W is the matrix of psi basis functions."
  • B-Spline -- from Wolfram MathWorld: "Define the basis functions as ..."
  • Green's Function--Helmholtz Differential Equation -- from Wolfram: "Define the basis functions phi_n as the solutions to the homogeneous Helmholtz differential equation ..."
  • NURBS Curve -- from Wolfram MathWorld: "... Ni,p are the B-spline basis functions, ..."
  • NURBS Surface -- from Wolfram MathWorld: "... and Nj,q are the B-spline basis functions, ..."
Search statistics
  • "basis function" finds 15700 pages
  • "basis function" without "radial basis function" finds 8080 pages
  • "basis function" with "numerical analysis" finds 191 pages
  • "basis function" with "quantum mechanics" finds 149 pages
  • "basis function" with "fourier analysis" finds 81 pages
  • "basis function" with "linear algebra" finds 75 pages
  • "basis function" with "functional analysis" finds 67 pages
  • "basis function" with "predictive analytics" finds 0 pages
Sample titles
  • Radial Basis Functions: Theory and Implementations "Radial basis function methods are the means to approximate the multivariate functions we wish to study in this book." (p 2)
  • Modelling And Identification With Rational Orthogonal Basis Functions "... using orthonormal infinite impulse response (IIR) filters as basis functions, much more efficient model structures can be obtained." (p 1)
  • An Introduction to Numerical Analysis "... the spline basis function ..." (p 309), "... finite element basis function ..." (p 394)
  • Mathematical Methods in Chemistry and Physics "... of sine and cosine functions which will be used as a function basis set." (p 67)
  • Group Theory and Quantum Mechanics "In this case we need two labels for a basis function, one for the irreducible representation and one for the row (or column) within the representation." (p 39)
  • Mapped Vector Basis Functions for Electromagnetic Integral Equations "... the majority of approaches in use rely on low-order polynomial basis functions, negating some of the accuracy advantages of curved cells." (p 1)
  • Wavelets for Sensing Technologies "The basis functions that span the subspace may form an orthogonal, a semiorthogonal, or a biorthogonal basis." (p 19)
  • An Introduction to Wavelets Through Linear Algebra "Figure 22 shows the graphs of a few 4th level real Shannon wavelet basis functions ..." (p 233)
  • Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements "Now if the set of basis functions \{\phi_i\}_{i=1}^N is chosen in such a way that they are orthogonal with respect to the energy inner product, then ..." (from the section on the Galerkin method, p 344)
Clearly the term is commonly used and deserves an article. The question is what are the subjects of the article? I don't have the expertise to characterize the subject areas any of these titles would fall into.
--Jtir 11:40, 25 September 2006 (UTC)

i'm kinda not surprised that combining "basis function" with a general subject area (say, "numerical analysis") doesn't land many matches. i would think most authors would assume their readers know the general subject area they're talking about. you might have better luck searching on "numerical analysis" with MSC numbers for that category like on ams.org/journals/ or, better yet, ams.org/mathscinet/.

i s'pose i would be curious, though, what would happen if you searched on "basis function" with "approximation/approximating/approximate/...", "finite element(s)", "finite volume(s)", "galerkin", "boundary integral", "wavelet", "fourier" (to catch "analysis", "transform", "series", and others), "banach", "sobolev", "lebesgue", and "hilbert". if i think of others, i'll let you know.

keep up the good work. Lunch 17:10, 27 September 2006 (UTC)

Thanks for your suggestions and encouragement. I am still refining the technique, so there are some obvious flaws, e.g. these searches may find the same title more than once, and some titles have articles by several authors, any of whom may use only one of the terms. The labor intensive part is skimming titles for possible quotes, running a search on the title at amazon.com where the full text is available, skimming it, and transcribing the quote (I have manually entered all or part of a few quotes). --Jtir 19:27, 28 September 2006 (UTC)

[edit] Further comments

in my humble experience, "basis function" is acceptable usage akin to "basis vector" when applied to a vector space of functions. "eigenfunction" is also used in place of "eigenvector".
in numerical analysis we often deal with separable Banach spaces which almost always have a Schauder basis. (if a normed space has a schauder basis, then it's separable, but the converse isn't true; it seems, though, that counterexamples are "pathological" or not commonly encountered.) as for a functional analysis reference, kreyszig's "introductory function analysis with applications" is a common one. i believe "eigenfunction" and "basis function" appear in it. (with a quick glance, i found "eigenfunction"; unfortunately, the book isn't concerned with bases much at all so i haven't found "basis function" yet, but i'm looking.)
i think the example given here was by someone thinking of the fourier transform as applied to L2 which has a Schauder basis. (and in the Schauder basis article, this is indeed used as an example.)
maybe this article needs to be merged with one of the other appropriate articles. Lunch 15:44, 25 September 2006 (UTC)
i s'pose i should add i mentioned the schauder basis for L^2 because it's in the article. not all function spaces are infinite dimensional (duh :). drawing from numerical analysis, we often deal with (finite dimensional) spaces of polynomials. in one dimension, there are several bases one could use such as the canonical basis (1,t,t^2,...,t^n), the Chebyshev basis, the Lagrange basis at a given set of knots, and a whole host of others. i don't think i could bring myself to say t^2 is a basis vector in the canonical basis; it's much more natural to say "basis function."
it looks like all of the articles cited above from the AMS MathSciNet search and some of the MathWorld articles are along these lines (from numerical analysis). some of the other MathWorld articles are about fourier analysis. maybe the proper category is "analysis" and not solely "functional analysis". Lunch 16:04, 25 September 2006 (UTC)
Thanks -- very helpful. I've added Kreyszig to the not using list, because the term basis function is not in the index reproduced at www.amazon.com. If you find the term in the text, please move the reference. --Jtir 16:59, 25 September 2006 (UTC)

[edit] A list of related articles (feel free to add):

[edit] A list of linked articles

These articles link to Basis function. User and talk page links were removed. The classification is only approximate, however it shows that not just math articles use the term Basis function. A wikipedia search finds many other examples of the term being used. --Jtir 21:02, 25 September 2006 (UTC)

Math

Physics

Chemistry

Engineering/Technology

Business/Finance

Redirects

Lists

[edit] lead sentence and categories

I changed analysis to numerical analysis in the lead sentence, so that it now begins "In mathematics, particularly numerical analysis, ....", because there is strong evidence that basis function is used in numerical analysis and the lead is now more specific. It might be worth noting somewhere that radial basis function is sometimes abbreviated to basis function. Further, numerical analysis is not a subdivision of analysis according to the mathematical analysis article.

We need to identify other areas. Lunch previously noted that "some of the other MathWorld articles are about fourier analysis." I'm not sure which ones ... maybe Green's Function--Helmholtz Differential Equation?

I also removed the functional analysis category because there is no evidence whatsoever that basis function is used in that subject. I am now zeroing in on linear algebra.  :-) --Jtir 12:36, 26 September 2006 (UTC)

Sure, the change from "analysis" to "numerical analysis" sounds good. Something about it makes me uncomfortable, but I can't put my finger on it at the moment.
With regards to numerical analysis not being part of analysis, that seems to be an oversight in the analysis article. I'll change it. Note that the first sentence says, "Analysis is a branch of mathematics that depends upon the concepts of limits and convergence." Well, loosely speaking, numerical analysis is the study of convergence of approximations of a quantity of interest.
With regards to "radial basis function" being shortened to "basis function," I think that's only done in context. That is, an author may get tired of saying "radial basis function" over and over again, and just shorten it out of convenience.
With regards to the MathWorld articles, I originally meant to say "the Google books results." Oops. But now that you mention that MathWorld article on the Helmholtz equation, they're using the term "basis function" to mean "eigenfunction of the Laplace operator which we can then use as a basis for expanding other functions." (That is, in an analogy to the finite dimensional case, a diagonalizable operator has a complete set of eigenvectors (that can be used to make their own coordinate system).)
With regards to the "linear algebra" category, I was just going cat crazy. You might lop it off since there's the more specific category "numerical linear algebra." But I don't know that perhaps other algebraists might use the term which is why I included it originally. That is, chances are someone, somewhere in all the fields I originally put in has used the term... Lunch 19:00, 26 September 2006 (UTC)

[edit] Discussion from Wikipedia_talk:WikiProject_Mathematics

There seems to be disagreement over whether the term basis function is used in functional analysis. I don't know enough about the subject to have an opinion. Could someone comment at Talk:Basis function? --Jtir 13:04, 25 September 2006 (UTC)

There is a problem with the weakness of the article. There must be several areas, eg wavelets, where this is a relevant concept. Charles Matthews 13:12, 25 September 2006 (UTC)
Correct. I looked at the what links here list and found wavelets, plus articles in chemistry, physics, engineering, and business that link to Basis function (I've put a culled, classified, and alphabetized list of linked articles at Talk:Basis_function). A wikipedia search finds many other examples of the term being used. It is starting to seem to me that making the article a dab would be preferable to trying explain all possible uses of the term. I don't have enough WP experience, though, to know what the implications are. --Jtir 21:26, 25 September 2006 (UTC)
A dab page makes mainly sense if we have separate articles on different meanings of the words "basis function". In mathematical use, isn't there a common meaning: an element of some basis of a vector space whose elements are functions? The main problem of the article may be that it starts with the words "In functional analysis" instead of "In mathematics". --LambiamTalk 22:38, 25 September 2006 (UTC)
an intro sentence might be, "In mathematics -- particularly analysis -- a basis function is an element of the basis for a function space. The use of the term is analogous to basis vector for a vector space." (NB: some of those words are dab pages so the links are Analysis (mathematics) and Basis (linear algebra).) Lunch 00:56, 26 September 2006 (UTC)
With this formulation, couldn't all the technical content of the article be removed? Basically the article is saying that basis function is a synonym for basis vector in some usages. If so, the article could become a redirect to basis (?) which could parenthetically note the same thing. --Jtir 16:10, 26 September 2006 (UTC)

I don't think a simple redirect to Basis is a good idea. When dealing with bases in function spaces, a Hamel basis (which is what that page focuses on) is usually not the tool of choice. Instead one typically deals with a Schauder basis or, in the more specific Hilbert space setting, an orthonormal basis. Sometimes the word is stretched a bit, such as in the context of Riesz basis (which I think is really just a frame). Michael Kinyon 16:20, 26 September 2006 (UTC)

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