Talk:Basis function

message: This page should give a definition of a basis function first, bolding the key word. Fresheneesz 05:45, 23 March 2006 (UTC)
replies: text: ...and you shouldn't define something with itself.  —Preceding unsigned comment added by 66.226.212.190 (talk) 15:53, 29 November 2010 (UTC)
thread_title: definition

message: 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)
thread_title:  Square integrable 

message: 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)
thread_title:  a comment 

message: 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)
thread_title:  Basis of a vector space 

message: :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: "... 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: "... N_{i,p} are the B-spline basis functions, ..."
:::*NURBS Surface -- from Wolfram MathWorld: "... and N_{j,q} are the B-spline basis functions, ..."
::*http://books.google.com/
:::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)
replies: text: 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)
thread_title: References using the term ''basis function''

message: 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)
thread_title: A list of linked articles

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Discussion from [[Wikipedia_talk:WikiProject_Mathematics]] 

message: Sorry, I saw this on the copy-edit list and thought it just needed some punctuation and phrasing cleaned up.  Did a bit and then discovered it was a can of worms by reading the discussion!  Have learned to look at discussion first. Prufi 21:27, 3 January 2007 (UTC)
thread_title:  sorry, tried to copy-edit 

message: The section "Fourier Basis" don't tell what a Furier Basis is!! I supose The base of L^2(0,1) cited in the section is a Fourier Basis, but it should be explicicited.  — Preceding unsigned comment added by Nadapez (talk • contribs) 23:01, 8 August 2013 (UTC)
thread_title:  Fourier Basis?