Proving Orthogonal Polynomials w/ Respect to Measure w(x) & Matrix M

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Discussion Overview

The discussion revolves around the proof of the existence of a Hermitian matrix M related to orthogonal polynomials defined with respect to a measure w(x). Participants explore the relationship between the orthogonal polynomials and the matrix representation, questioning the validity and clarity of the initial equation presented.

Discussion Character

  • Exploratory, Debate/contested, Conceptual clarification

Main Points Raised

  • One participant presents an equation involving orthogonal polynomials and a Hermitian matrix, seeking to establish a connection between them.
  • Another participant expresses confusion regarding the equation and requests a specific example of an orthogonal system and its corresponding matrix.
  • A third participant suggests a dismissive approach to proving the claim, implying that one could write a proof without verifying its correctness.
  • A fourth participant criticizes the reaction to dissenting opinions, suggesting that the group is not genuinely pursuing truth.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, with multiple competing views and disagreements on the clarity and validity of the initial claim and the responses to it.

Contextual Notes

Some assumptions regarding the definitions of the orthogonal polynomials and the measure w(x) are not explicitly stated, which may affect the interpretation of the initial equation. The discussion also reflects varying levels of engagement and openness to differing viewpoints.

zetafunction
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given a set of orthogonal polynomials with respect to a certain measure w(x)

[tex]\int_{a}^{b}dx w(x) P_{n} (x)P_{m} (x) = \delta _{n,m}h_{n}[/tex]

how can anybody prove that exists a certain M+M Hermitian matrix so

[tex]P_{m} (x)= < Det(1-xM)>[/tex] here <x> means average or expected value of 'x'

if we knew the set of orthogonal polynomials [tex]P_{m} (x)[/tex] for every 'm' and the measure w(x) , could we get the expression for the matrix M ??
 
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Your equation doesn't make much sense to me. How about providing an example. A particular orthogonal system and the corresponding matrix.
 
mathaino said:
I recommend "proof by believe", i.e. write something that looks like a proof, believe it is a correct proof, without being able to check whether it is correct.

Looks like we got ourselves a troll.
 
Interesting way to react to posts that do not tickle the own ears - delete them. You guys are not seekers of truth. Ibn al Haytham would be ashamed for you all.
 

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