# Questions tagged [orthogonal-polynomials]

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

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### Hermite Polynomial and its Expectation

Currently, I'm stuck to some statement in a paper (in chapter 8: Nonlinear Model, from page 26 ~27). Although this topic generally covers statistics and machine learning theory, my main question is ...
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### Lagrange interpolation and orthogonal polynomials

Suppose that $\{p_i(x)\}_{i=0}^{n}$ are pairwise orthogonal polynomials on the interval $[a,b]$, It means, $$\int_{a}^{b} p_i(x)p_j(x)dx = 0\ , \;{i\neq j}$$ wherein $p_i(x)$ for all $i$ is a ...
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### Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?

After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations: with $M$ the KummerM confluent hypergeometric funcion and ...
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### Largest root of a linear combination of Chebyshev polynomials

I wonder if we can say something about roots of a linear combination of Chebyshev polynomials of the first kind. I have an example in my hand: $$(m+1)T_n(x)+(m-3)T_{n-2}(x)=0$$ for some $m>0$. I ...
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### Construction of orthogonal matrices from orthogonal polynomials

Let $\{P_n\}_{0\le n\le N}$ be the family of orthogonal polynomials associated with the following inner product: $$\langle f, g \rangle_{1} = \sum_{k=0}^{N}{f(k)g(k)w_1(k)}$$ where $w_1(\cdot)$ is ...
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### Simplify this sum of Bell polynomials similar to the generating function

During my research, I encountered this expression: $$\sum_{p=0}^{\infty} \frac{1}{(p+1)!} \sum_{k = 0}^{p} k! \, u^k \, B_{p,k}(f_1, \dots, f_{p-k+1})$$ with the arguments of $B_{p,k}$ being the ...
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### Charlier-Sobolev-type orthogonal polynomials

I am currently reading an article, and I am a bit stuck on understanding a sentence in this article. The sentence, as indicated in the image, is: " In the case of Charlier polynomials, the ...
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### Biorthogonal sequence for monomials in L2(0,1)

A sequence $(x_n^*)_{n \in \mathbb{N}}$ is a biorthogonal sequence for a sequence $(x_n)_{n\in\mathbb{N}}$ in a Hilbertspace $H$ if $$(x_m^*, x_n) = \delta_{mn}$$ for all $m,n \in \mathbb{N}$. Does ...
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### Kernel Polynomials and Extremality

I am trying to prove the following theorem: Theorem Let $x_0 \in \mathbb{R}$ and $q_n(x)$ a polynomial of degree at most n, normalized by the following condition: $\int_a^b (q_n(x))^2 w (x)dx$=1. The ...
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### Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform.

Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction $$\frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$...
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### Proving the reproducing property of kernel polynomials

I need to prove the following property related to kernel polynomials: $\int_a^b K_n(t,x)q_n(x)w(x)dx=q_n(t)$, where $q_n(x)$ is a polynomial of degree less or equal to $n$, $w(x)$ is a weight function ...
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### Finding the minimum value of an integral using least squares-mean function approximation

I am starting to study some concepts related to orthogonal polynomials and my teacher told me to prove the following theorem, Theorem The integral $\int_{a}^{b} Q_n^2(x)w(x) dx$ where $Q_n(x)$ is any ...
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### A bound for error of least square

I'm approximating $f(t)$ by $P_n(t)$, a real polynomial of arbitrary degree $n$, by minimizing the $L_2$ norm of its difference in $[0,1]$ $$R_n=|| f(t)-P_n(t)||_2^2=\int_0^1(f(t)-P_n(t))^2\,dt$$ ...
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### Implication of dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L_2[a,b,p]$, where $Q_n(x)$ is a polynomial of $n$-th degree.

Dot product $(Q_n(x), 1)\equiv \int_a^b {p(x)Q_n(x)dx} = 0$ in $L_2[a,b,p]$, where $Q_n(x)$ is a polynomial of $n$-th degree. According to my textbook this happens only if $Q_n(x)$ has at least one ...
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### Jacobi Orthogonal Polynomials - Scipy (Python) vs My Recurrence Implementation

I implemented the Jacobi orthogonal polynomial recurrence relation using the following: Source: http://lsec.cc.ac.cn/~hyu/teaching/shonm2013/STWchap3.2p.pdf However, when I compare to the Scipy (...
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### Proving that the composition of orthogonal transformations is orthogonal in a finite dimensional inner product space $V$.

Given $S$,$T$ orthogonal transformations in an inner product space $V$ over $\mathbb R$ such that $\dim(V) < \infty$. Also given that $v \in V$ | ($S$$\circ$$T) (v) = S(T(v))$ I need to prove that ...
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