All Questions
Tagged with legendre-polynomials functional-analysis
19
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Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 993
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89
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Generating function of orthogonal polynomial basis
I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function ...
- 323
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64
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Comparing the different bases for representing a function
Suppose I have some function $f(x)$.
I know that this function can be represented in several different bases. For example, we can express this using Legendre polynomials, Hermite polynomials, Fourier ...
1
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1
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How $x^n$ is linearly represented by Legendre polynomials
I recently come across a problem with respect to Legendre polynomial as follows.
Let $L^2[-1,1]$ be the Hilbert space of real valued square integrable functions on $[-1,1]$ equipped with the norm $\|f\...
- 493
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1
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480
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Orthonormal polynomial basis of $L^2([0,1])$
I was wondering if, given a natural number $i\in \mathbb N$, there exists an orthonormal basis (w.r.t. the standard scalar product) $(p_n)_{n \in \mathbb N}$ of $L^2([0,1])$ such that $p_n$ is a ...
- 1
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1
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212
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Series involving product of Legendre polynomials
I need to compute the following sum:
$$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$
where $P_n(x)$ are the Legendre polynomials. Can anyone help me?
- 33
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Determining whether a singular endpoint is of limit-circle or limit-point case.
The ODE is $−[(1−x^2)u′]′-\mu u=f(x)$ over the interval $[-1,1]$.
I want to determine whether the singular endpoint $x=1$ is of a limit circle or limit point type. To do so, we were given a Theorem ...
- 1,019
2
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504
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Function expansion in a basis of associated Legendre polynomimals
The Legendre polynomials $P_l(x)$ are complete in that any continuous function on $[-1,1]$ can be expanded as,
$$
f(x) = \sum_{l=0}^{\infty} a_l P_l(x)
$$
(see here for example). However, what is the ...
- 1,026
2
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159
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Decay rate of the coefficients in the Legendre series expansion
Let $u$ be a function in $L^2_{\nu}([0,\pi])$ where $\nu=2\sin(x)$
Let $\hat{u}_n$ denote the truncated Legendre series expansion of $u$ defined by
\begin{equation*}
\hat{u}_n:= \...
- 311
1
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520
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Why does $\int_{-1}^1 ((1-x^2)(P_m'P_n-P_n'P_m))'\,dx = 0$ for Legendre polynomials?
I was looking at a proof of the orthogonality of the Legendre polynomials in Lebedev's Special Functions and their Applications:
I can't understand why the integral of the first term vanishes. I ...
- 8,728
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54
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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 $$\frac{\|{x}\|_2}{2} \leq \...
- 31
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1
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380
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Since the solution of the Legendre’s equations are the Legendre polynomials $P_n(\mu)$, write the general solution for $u(r, \theta)$.
I am working on the problem
Consider the steady-state of the heat equation in a ball of radius a centred at the origin. In spherical coordinates, the ball occupied the region $0 \le r \le a$, $0 \...
user575678
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654
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Showing that the first two Legendre polynomials, $P_0(x) = 1$ and $P_1(x) = x$ are orthogonal for $x \in [−1,1]$
I would like assistance with the following problem:
Show that the first two Legendre polynomials, $P_0(x) = 1$ and $P_1(x) = x$ are orthogonal for $x \in [−1,1]$.
Determine constants $\alpha$ and $\...
- 4,322
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Summation of Legendre polynomials with two power functions
I know there is a generating function of Legendre polynomials, that is
$$g(x,t)=1/\sqrt{t^2-2tx+1}=\sum _{n=0}^{\infty }t^n P_n(x),$$when $ \left| t\right| <1 $.
So is there any expression in ...
- 41
2
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1
answer
478
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Minimal orthogonal projections, and distances in Hilbert spaces.
Let $X=L^2\left([-1,1]\right)$ be the inner product space of Lebesgue integrable functions from $[-1,1]$. And denote $\mathcal{P}_{n-1} \subset X$ the subspace of polynomials with a degree not greater ...
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