Talk:Basis function

definition

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

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)

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/

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:

http://www.ams.org/journals/ (search was for titles using the term)

[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.

http://mathworld.wolfram.com/

Haar Function -- from Wolfram MathWorld
B-Spline -- from Wolfram MathWorld
Green's Function--Helmholtz Differential Equation -- from Wolfram ...
NURBS Curve -- from Wolfram MathWorld
NURBS Surface -- from Wolfram MathWorld

http://books.google.com/

Radial Basis Function Neural Networks With Sequential Learning - Page vii
Modelling And Identification With Rational Orthogonal Basis Functions - Page xxiii
Mathematical Methods in Chemistry and Physics - Page 67 "... of sine and cosine functions which will be used as a function basis set."
Group Theory and Quantum Mechanics - Page 39 "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."
Mapped Vector Basis Functions for Electromagnetic Integral Equations - Page 29

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

Basis (linear algebra) aka Hamel basis
Schauder basis
Topological vector space
Function space
Hilbert space for the fourier analysis example
functional analysis, numerical analysis, Fourier analysis, and analysis (mathematics)