All Questions
Tagged with legendre-polynomials linear-algebra
16
questions
0
votes
1
answer
68
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Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors? [closed]
I was not able to find resources online and was wondering so since it would greatly help me with my work if I can directly apply what I have learned from linear algebra.
- 17
1
vote
1
answer
267
views
Uniqueness of the nodes for Gauss-Legendre quadrature
Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$.
Wikipedia says that
This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ...
user1095295
4
votes
1
answer
177
views
The values of $P_n(x)$ at the zeros of $P'_n(x)$
I recently come across a problem with respect to Legendre polynomial as follow.
For any $n \in \mathbb{N}$, $P_n(x) := \frac{1}{2^n n!}\frac{{\rm d}^n (x^2-1)^n}{ {\rm d} x^n} $ is the classical ...
- 493
2
votes
1
answer
342
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Coefficient of the Highest Degree in the Power Series Solution to Legendre's Differential Equation
The Legendre's differential equation
$$(1-x^2)y''-2xy'+n(n+1)y=0$$
Substitute $y=\sum_{m=0}^\infty a_mx^m$
$$
\sum_{m=2}^\infty m(m-1)a_mx^{m-2}-\sum_{m=2} ^\infty m(m-1)a_mx^m-\sum_{m=1}^\infty ...
- 7,674
3
votes
0
answers
54
views
Solving an inequality containing Legendre polynomials
Let it be given that $A$ is a real valued $m$ by $n$ matrix with entries $p_j\left(\frac{i}{m-1}\right)_{i,j=0}^{m-1,n-1}$, where $p_j$ is a Legendre polynomial. Show that $$\frac{\|{x}\|_2}{2} \leq \...
- 31
1
vote
1
answer
250
views
Fourier-Legendre Series and Closure
My professor taught us this last week
Legendre Polynomials form a maximal orthogonal set in this larger
space (previously we were looking at polynomials)
This means that there is no nonzero ...
- 697
1
vote
1
answer
2k
views
Legendre Polynomials form a Basis
How can I show that Legendre Polynomials form a basis for all polynomials?
I think it should be enough to prove that they are linearly independent but how exactly should I do that?
- 49
0
votes
1
answer
118
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basis for polynomials
A basis for the polynomials that shows up in physics are the “Legendre polynomials.” The first few, B = {1, x, ((3/2)x^2)-(1/2)}, are a basis for P2. Calculate [3-x+x^2]B.
I understand a basis is the ...
- 819
2
votes
2
answers
339
views
Intuition for polynomial bases
In my linear algebra course I stumbled upon the following observations.
We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$.
$f(x)$ may be composed of elementary functions or not, but in ...
- 21
3
votes
4
answers
3k
views
How to express $2x^3-x^2-3x+2$ as a linear combination of Legendre polynomials
I have used the formula
\begin{align}p_0(x)&=1\\
p_1(x)&=x\\
p_2(x)&=\frac12(3x^2−1)\\
p_3(x)&=\frac12(5x^3−3x)
\end{align}
$$2x^3-x^2-3x+2=Ap_3(x)+Bp_2(x)+Cp_1(x)+Dp_0(x)$$
EDIT-
\...
- 81
5
votes
0
answers
387
views
Pointwise convergence of a Legendre polynomial expansion
$\langle\cdot,\cdot\rangle$ is the dot product on the real vector space $\mathcal C ([0,1],\mathbb R)$ defined by $\langle f,g\rangle = \int_{-1}^1 fg$, and $(L_n)$ is the family of normalised ...
- 435
1
vote
1
answer
1k
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Vector Space Spanned by Legendre Polynomials
Problem
Let $V$ be the vector space over $\mathbb{R}$ spanned by Legendre Polynomials:
$P_0(x)=1$,$\space\space$$P_1(x)=x$,$\space\space$$P_2(x)=\frac{1}{2}(3x^2-1)$,$\space\space$$P_3(x)=\frac{1}{...
- 135
1
vote
1
answer
121
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can Legendre polynomials take on different forms?
My prof. assigned some homework that has us compute legendre polynomials, but I'm getting polynomials that are different from ones that I reference with on like Wolfram and Wikipedia. I think the ...
- 1
3
votes
1
answer
77
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Showing $y_1$ or $y_2$ are not polynomials
proof that $y_1$ or $y_2$ are not a polynomial for any $n$
$$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$
$$ y_2(x)=x-\frac{(n-1)(n+2)}{3!}x^3+\frac{(n-3)(n-1)(n+2)(n+4)}{...
1
vote
1
answer
895
views
projection of inner products
Update of question
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define our inner product to be:
$$
\langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(...
- 105