Basis functions Definition and 18 Threads

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).

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  1. pellis

    I Calculating group representation matrices from basis vector/function

    Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...
  2. M

    A Finding eigenvalues with spectral technique: basis functions fail

    Hi PF! I'm trying to find the eigenvalues of this ODE $$y''(x) + \lambda y = 0 : u(0)=u(1)=0$$ by using the basis functions ##\phi_i = (1-x)x^i : i=1,2,3...n## and taking inner products to formulate the matrix equation $$A_{ij} = \int_0^1 \phi_i'' \phi_j \, dx\\ B_{ij} = \int_0^1...
  3. M

    A Basis functions and spanning a solution space

    Hi PF Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation $$\langle...
  4. M

    Mathematica Can I Scale Down Basis Functions Without Losing Zero Force?

    Hi PF! I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...
  5. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

    I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...
  6. J

    A Linear Regression with Non Linear Basis Functions

    So I am currently learning some regression techniques for my research and have been reading a text that describes linear regression in terms of basis functions. I got linear basis functions down and no exactly how to get there because I saw this a lot in my undergrad basically, in matrix...
  7. L

    Info on Bessel functions & their use as basis functions.

    Hello all,As an exercise my research mentor assigned me to solve the following set of equations for the constants a, b, and c at the bottom. The function f(r) should be a basis function for a cylindrical geometry with boundary conditions such that the value of J is 0 at the ends of the cylinder...
  8. A

    1000 Basis functions limitation - Gaussian 09?

    Does anyone know if there is a limitation of the number of basis functions for freq calculation in Gaussian 09? I have 2 freq calculations, one has 879 basis functions and the other one has 1047 basis functions. Gaussian 09 couldn't finish the 1047 basis function calculation no matter how...
  9. mnb96

    Orthonormal basis functions for L^2(R)

    Hello, are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)? According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of...
  10. R

    Basis functions of a differential equation, given boundary conditions

    First off, I've never taken a differential equations class. This is for my Math Methods for Physicists class, and we are on the topic of DE. Unfortunately, we didn't cover this much, so most of what I am about to show you comes from the professor giving me tips and my own common sense. I'd...
  11. A

    Sobolev type norms and basis functions

    Hello everybody, I am given a "Sobolev type innerproduct" \langle f,g \rangle_{\alpha} = \langle f,g \rangle_{L^2} + \alpha \langle Rf,Rg \rangle_{L^2} for some \alpha \geq 0 and R some differential operator (e.g. the second-derivative operator). My question is now whether a function...
  12. S

    How to Find the B Matrix in Matlab for Wavelet Basis Functions?

    Hi all, Say that I have a 1D signal such that f=Bw where f is the signal B is the basis functions and w is the wave co-efficients. The question that I have is how do I find the B matrix in Matlab. I am looking through WaveLab and Rice Wavelet packages but simply cannot find an answer. As...
  13. S

    Finding basis functions for approximating transcendental function

    I am working on a problem where I want to approximate a transcendental function of the form f(x) = x^Ne^{x} for x \geq 0 as a linear combination of functions of the form x^v \text{where} -1 < v < 0. How can I find the basis functions of the desired form to represent my transcendental...
  14. F

    Solving Basis Functions Homework w/ Constants A_n & B_n

    Homework Statement Given x in the interval [0, \pi], let \phi_{0}(x) = 1, and \Phi_{n} (x) = sin ((2n-1)x). Show that there are constants: {A_{n}}^{n=0}_{\infty} and {B_{n}}^{n=0}_{\infty} such that: \sum^{n=0}_{\infty}A_{n}\phi_{n}=\sum^{n=0}_{\infty}B_{n}\phi_{n} But A_{n}...
  15. Somefantastik

    Basis functions for polynomial

    Homework Statement For I = [a,b], define: P3(I) = {v: v is a polynomial of degree ≤ 3 on I, i.e., v has the form v(x) = a3x3 + a2x2 + a1x + a0}. How to show v is uniquely determined by v(a), v'(a), v(b), v'(b). Homework Equations The Attempt at a Solution I'm not exactly sure...
  16. Dale

    Maxwell's Eqns: EM Basis Functions Explored

    Are there any sets of basis functions that are particularly useful for Maxwell's equations? I was thinking about Fourier just because it is the first basis I always think of, but I don't know that it would actually be a convenient basis. For example, I don't know that curl or divergence would...
  17. mnb96

    Fourier Basis functions question

    Hi, the continuous Fourier Transform is often defined on a finite interval, usually [-\pi,\pi]: \hat{f_k} = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}dx If I understood correctly, this allows the basis functions to be defined so that they have norm=1, and they form an orthonormal basis for...
  18. D

    Wave function in terms of Basis Functions

    Problem We have the function g(x)=x(x-a) \cdot e^{ikx}. Express g(x) in the form \sum_{n=1}^\infty a_n \psi_n (x) where \psi_n = \sqrt{\frac{2}{a}} \sin \(\frac{n\pi x}{a}\) Solution I have absolutely no clue as to how to start... I know a bit about Fourier series, but here, the...
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