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

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In summary, the conversation discusses the existence of a Hermitian matrix M that relates to a set of orthogonal polynomials P_m(x) with respect to a measure w(x). The equation given suggests a possible expression for the matrix M, but without further information or an example, it cannot be proven. The idea of a "proof by believe" is also mentioned, but it is not a reliable method for finding true proofs.
  • #1
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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  • #2
Your equation doesn't make much sense to me. How about providing an example. A particular orthogonal system and the corresponding matrix.
 
  • #3
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.
 
  • #4
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.
 

1. What are orthogonal polynomials?

Orthogonal polynomials are polynomials that are defined with respect to a specific weight function, such that the inner product of any two polynomials is equal to zero. This means that the polynomials are perpendicular to each other in a mathematical sense.

2. How is orthogonality determined in orthogonal polynomials?

The orthogonality of polynomials is determined by the weight function w(x) and the associated inner product. If the inner product of two polynomials is equal to zero, then they are considered orthogonal with respect to the weight function.

3. What is the role of the weight function in orthogonal polynomials?

The weight function, denoted as w(x), plays a crucial role in defining the inner product and determining the orthogonality of polynomials. It is a non-negative function that assigns a weight to each point on the x-axis, and it is an essential component in calculating integrals and determining the properties of orthogonal polynomials.

4. How are orthogonal polynomials used in scientific research?

Orthogonal polynomials have many applications in various fields of science, including physics, engineering, and statistics. They are commonly used in numerical analysis and approximation techniques, as well as in solving differential equations and performing data analysis.

5. What is the significance of proving orthogonal polynomials with respect to measure w(x) and matrix M?

The proof of orthogonality with respect to a weight function and matrix is essential because it confirms the validity of the polynomials and their properties. It also allows for the use of various mathematical techniques and tools to analyze and manipulate the polynomials for practical applications.

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