Questions pertaining to certain sets of polynomials that satisfy an orthogonality criterion with respect to some specified inner product.

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32 views

Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
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0answers
17 views

Example of incomplete orthogonal system in L2 space

I was asked to provide an example of an incomplete orthogonal in any L2 space and was wondering if this is a valid answer. Take some known complete system, for example the Hermite polynomials over ...
2
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0answers
29 views

Gaps between roots of trigonometric polynomials?

Given a polynomial in $e^{\mathrm{i}k t}$ of the form $$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$ with $\bar c_{-k} = c_k$, is there a good way of characterising how close its roots can ...
5
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1answer
37 views

Asymptotics bound on Jacobi polynomials in the complex plane and for large $n$

Dear mathematicians and theoretical physicists, I am a theoretical physicist and I am bothering to you since I need to know some asymptotic and analytical properties of Jacobi polynomials ...
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0answers
24 views

Can an orthognal vector be represented by a linear combination of nonorthognal basis?

Given a orthogonal set of vectors g_i(x) can they be represented as: $g_i(x) = \sum c_i \phi_i(x)$ Where $\phi(x)$ meet the support requirements of g_i(x) but are not orthogonal.
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1answer
36 views

Proof that Qm(x) and Pn(x) have the same sign

I have difficulty in this exercise, is from the book Apostol Calculus Vol 2, I really want to know is there is a technique that can show that the two polynomials share the same roots. Thanks.
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0answers
63 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
5
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2answers
263 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
1
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1answer
24 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
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1answer
61 views

How does one prove Rodrigues' formula for Legendre Polynomials?

I am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n. I've been told that the proof uses complex analysis, of ...
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0answers
13 views

Best uniforme approximation of nule function in the meaning of Tchebychev

I would be interest to know , why exactly approximate a nule function and it is in the same time nule ? I would be like someone give me enough (papers, link ...) about "The best uniforme ...
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2answers
30 views

not complete polynomial and roots

It might be very simple but I need a formal proof for accepting or rejecting the idea below. Let g be a polynomial of the order of n given below $$ g(L)=1-\theta_1 L-\theta_2 L^2- \ldots -\theta_n ...
11
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1answer
133 views

Will this sequence of polynomials converge to a Hermite polynomial pointwise?

While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions ...
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2answers
62 views

Legendre polynomial expansion of the unit step function.

The problem is to determine the expansion of the unit step function in terms of Legendre polynomials on the interval $[-1,1]$. Here the Legendre polynomials are the family of orthogonal polynomials ...
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0answers
18 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
0
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0answers
29 views

Laplace transform and Laguerre Polynomials

let be the Laplace transform of a function $$ F(s)= \int_{0}^{\infty}dtf(t)e^{-st} $$ then if the function satisfies that for every integer 'n' $ D^{n}f(x)=0 $ then $$ s^{n}F(s)= ...
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0answers
15 views

orthogonal polynomials and quadrature formulae

is it true that for every set of orthogonal polynomials $$ \int_{a}^{b} w(x)P_{n}(x)P_{m}(y) =a_{m,n}\delta _{m}^{n}$$ with respect a measure $ w(x)$ can we always find a quadrature formula for ...
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0answers
21 views

Finding another solution in certain form (confluent hypergeometric series)

Consider the differential equation $$xf''(x)+(\alpha+1-x)f'(x)+nf(x)=0 ..........(*)$$, we know that the Laguerre Polynomial ...
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0answers
24 views

Checking a solution of differential equation

I am trying to prove that the Laguerre polynomial $$L=L_n^{\alpha}(x)=\frac{x^{-\alpha}e^{-x}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+\alpha})= ...
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1answer
64 views

Fermat Last theorem on Poly-Euler numbers

The poly-Euler numbers, denoted as $E_{n}^{(k)}$, are defined by the following generating functions :$${2\operatorname{Li}_k(1-e^{-x}) \over 1+e^{-x}}=\sum_{n=0}^\infty E_n^{(k)}{x^n\over n!}$$ The ...
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1answer
33 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
3
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1answer
62 views

Numerical evaluation of polynomials in Chebyshev basis

I have high order (15 and higher) polynomials defined in Chebyshev basis and need to evaluate them (for plotting) on some intervals inside the canonical interval $[1,\,-1]$. A good accuracy near 1 and ...
2
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1answer
31 views

Orthogonality of Lagrange Polynomials in Hermite Inner Product

My question is as shown above. I have churned through the first part, but I am stuck on showing the orthogonality of the Lagrange polynomials. My first hope was to use the fact that the Lagrange ...
2
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0answers
46 views

Chebyshev polynomials property

I was wondering, is there a property of Chebyshev polynomials that allows to express $T_{n}(\lambda x)$ in terms of $T_{n}(x)$, for some $\lambda\in\mathbb{R}$? (where $T_{n}(x)$ denotes $n$-th order ...
4
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1answer
210 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
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0answers
42 views

Turning a summation into an integral

I have a summation of the form: $$y(x) = \sum\limits_{h=-L}^L\frac{A(h)\cdot R(h)^2}{((x-h)^2+R(h)^2)^{3/2}}$$ Where I wish to solve/optimise $R(h)$ (leaving $A(h) = const/h$) or $R(h)$ and $A(h)$ ...
0
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1answer
34 views

How could I prove that $P_n (1)=1=-1$ for the Legendre polynomials?

I could "prove" that $P_n(1)=1=-1$. But I'm not sure where is my mistake. If I use the generating function: $$\frac{1}{\sqrt{1-2z+z^2}}=\sum_{n=0}^{\infty}P_n(1)z^n$$ Since: $\sqrt{1-2z+z^2 } = ...
0
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0answers
24 views

a question about Jacobi polynomials

Imagine if I have a defined function $\omega(\alpha, \beta, \gamma)$, where $0<\alpha<2\pi$, $0<\beta<\pi$, and $0<\gamma<2\pi$. I can then expand this function into series just like ...
0
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1answer
25 views

Expanding unity in terms of orthogonal functions cos( alpha(i) * y)

It is written in the book I am reading without proof that if we expand unity in terms of orthogonal functions cos( alpha(i) * y), we get: (Please check this link) ...
1
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1answer
377 views

bijection between lattice path and permutations

I am studying Viennot's combinatorial model for the Laguerre polynomials this semester under the guidance of my math professor. If I understand correctly, a bijection exists between the number of ...
0
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1answer
42 views

Calculating Fourier expansion using Legendre Polynomials

I'm trying to write any function of the type $t^m$ using Legendre polynomials $P_n(t)$ . That means: $$t^m=\sum_{n=0}^\infty\langle P_n,t_m\rangle P_n =\sum_{n=0}^\infty a_{mn}P_n$$ Where I have to ...
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0answers
14 views

The coefficients of multivariate hermite polynomials

Can someone please help me with finding a table that contains the coefficients of the multivariate hermite polynomials? I have a really hard time understanding the subject and I hope such a table ...
0
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1answer
20 views

On Jacobi Polynomial

Prove that The $n$-th jacobi polynomial $P_n^{(\alpha, \beta)}(x)$ with parameter $(\alpha, \beta)$ defined by $P_n^{(\alpha, ...
3
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1answer
28 views

Constructing sequence of orthogonal polynomials for arbitrary scalarproduct?

I need to construct a sequence of orthogonal polynomial $(P_i)_{i=0}^{\infty}$ for a family of scalarproducts. I want to look at different scalar products $\langle P_n(x),P_m(x) ...
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0answers
56 views

Problems in Orthogonal Polynomials

Let $P_n (x) , n=0,1,...$ be the monic orthogonal polynomials associated to the weight function $w(x)$. Let $Q(x), n=0,1,2,...$ be the monic orthogonal polynomials associated to the weight function ...
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1answer
98 views

Hermite Polynomials Triple Product

Similar to the question Legendre Polynomials Triple Product, I would like to ask whether there are any explicit formulas for the inner product of the Hermite polynomial triple product \begin{align} ...
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0answers
21 views

Recurrence formula for orthogonal polynomials

Consider the recurrence formula: $P_n(x)=(x-c_n)P_{n-1}(x)-\lambda_n P_{n-2}(x)$ The problem consist on showing that $\xi_1<c_n<\eta_1$ where $[\xi,\eta]$ is the true interval of orthogonality ...
2
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1answer
116 views

Lie algebra (su(1,1)) from legendre polynomials; question regarding http://arxiv.org/abs/1205.6353

Apologies if this question is a duplicate. OK, so my question heavily involves the paper http://arxiv.org/abs/1205.6353 which nicely details the Lie algebra su(1,1) coming from the Laguerre $L_n$, ...
2
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1answer
50 views

Intuition behind the weight function

The inner product in a $L^2$ space can be defined as: $$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$ For Legendre polynomials, we define it as: $$\langle P_m,P_n\rangle =\int_0^1 ...
2
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0answers
78 views

Derive Differentiation Identity for Gegenbauer Polynomial

For $\lambda > 0$ and $n = 0,1,2,3,\ldots$, the following is an n-th order polynomial: $$ P_{\lambda,n}(x)=(1-x^{2})^{-\lambda+1/2}\frac{d^{n}}{dx^{n}}(1-x^{2})^{n+\lambda-1/2} $$ ...
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1answer
57 views

Equivalent definitions of Hermite polynomials

The most common definition of the (physicists') Hermite polynomials that I have found in the literature is the following: $$ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ Now, Wikipedia also ...
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0answers
13 views

Polynomials/Function family that remains within interval bounds

Is there such a thing as a family of orthogonal functions defined over an interval $x\in[a,b]$, where the output is the interval $f(x)=y\in[c,d]$ (e.g. for simplicity $[0,1] \rightarrow [0,1]$). I ...
3
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1answer
63 views

Why should we care about orthogonal polynomials?

I know the mathematical definition but I'm having a hard time understanding the utility of orthogonal polynomials. I'm not saying they are useless, far from that! It is just that I like understanding ...
3
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1answer
58 views

find the distance between 2 functions

f(x) = $e^x$. g(x) = $.5(e-1/e) + 3x/e$. How do you find||f-g||. The inner product is defined as $\int_{-1}^1 f(x)g(x) dx$. I've tried this: $\int_{-1}^1 (e^x - (.5(e- e^{-1} + 3x/e)))^2dx$. This ...
0
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1answer
47 views

find linear polynomial g that is closest to f, where $f(x) = e^x$ and the distance between the two

In the real linear space [-1,1] with inner product $\int^1_{-1} f(x)g(x)\,dx$. Find the linear polynomial $g$ nearest to $f$ and find $||g -f||^2$ for this $g$. My problem is that I simply don't ...
1
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1answer
77 views

Jacobi polynomials

We define the inner product on the space $\Bbb R[x]$ by $$\langle P,Q\rangle=\int_{-1}^1P(x)Q(x)(1-x^2)^\alpha dx$$ where $\alpha>-1$. I need to prove that for all $n\in\mathbb N$ ...
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0answers
33 views

approximating Gegenbauer polynomials (or ultraspherical or Jacobi)

Looking for hardcore orthogonal polynomial people here... If we hold the degree $\ell$ constant and take the order $\alpha$ to infinity, the Gegenbauer polynomial $G_\ell^{(\alpha)}$ approaches the ...
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2answers
188 views

Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$ T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right. $$ where ...
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2answers
96 views

Orthogonal polynomials on $[0,1]$

Are the orthogonal polynomials for the standard $L^2$ product on $[0,1]$ well-known? I couldn't find anything upon a quick web search.
2
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1answer
156 views

Orthogonal family of polynomials from derivative of Legendre polynomials.

Let $$f_n(x) = \frac{d^4}{dx^4}P_n(x)$$ where $P_n(x)$ are the Legendre polynomials. By which polynomial should one multiply this family to construct a family or orthogonal polynomials on $[-1,1]$ ...