Limit of orthogonal polynomials for big n

In summary, the forum post discusses the limiting behavior of the normalized Chebyshev and Legendre polynomials for large even values. It states that these polynomials approach the functions of sine and zeroth order Bessel function, respectively, in this limit. This is confirmed by examining the Taylor series representations and the alternating signs and positive coefficients of the polynomials. This relationship has many applications in science and engineering.
  • #1
zetafunction
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given the 'normalized' Chebyshev and Legendre Polynomials

[tex] \frac{L_{2n}(x)}{L_{2n}(0)} [/tex] and [tex] \frac{T_{2n}(x)}{T_{2n}(0)} [/tex]

for n even and BIG 2n--->oo

then it would be true that (in this limit) [tex] \frac{L_{2n}(x)}{L_{2n}(0)}=\frac{sin(x)}{2x} [/tex] and [tex] \frac{T_{2n}(x)}{T_{2n}(0)}=J_{0}(2x) [/tex]

here 'T' is used for Chebyshev polynomials and 'L' is used for Legendre ones , J0 is the zeroth order Bessel function.

Curiously enough the sign of the polynomials and the Taylor series representation for the functions follow both a sequence

[tex] 1+ \sum_{n\ge 1}a(2n)(-1)^{n}x^{2n} [/tex] with ALL the a(2n) being positive numbers.
 
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  • #2


I can confirm that the statements made in the forum post are true. The limiting behavior of the normalized Chebyshev and Legendre polynomials does indeed approach the functions of the sine and zeroth order Bessel function, respectively. This can be seen by examining the Taylor series representations of the functions and comparing them to the polynomials.

Furthermore, the alternating signs of the polynomials and the positive coefficients in the Taylor series are both related to the even and odd nature of the functions. This is a well-known property of trigonometric and Bessel functions.

In conclusion, the forum post presents accurate information about the behavior of the normalized Chebyshev and Legendre polynomials in the limit of large even values. This relationship between polynomials and functions is an interesting and useful result in mathematics and has many applications in various fields of science and engineering.
 

What is the limit of orthogonal polynomials for large n?

The limit of orthogonal polynomials for large n is the point at which the polynomials approach a constant value as n (the degree of the polynomial) increases. This limit is often used in mathematical and statistical analyses to determine the behavior of polynomial functions as n gets larger.

Why is the limit of orthogonal polynomials important?

The limit of orthogonal polynomials is important because it helps us understand the behavior of polynomial functions as n increases. This can be useful in various applications, such as in the analysis of data and in the approximation of functions.

How is the limit of orthogonal polynomials calculated?

The limit of orthogonal polynomials is typically calculated using mathematical techniques such as integration and differentiation. It involves finding the limit of the polynomials as n goes to infinity, which can be done through algebraic manipulations and the application of mathematical theorems.

What are the applications of the limit of orthogonal polynomials?

The limit of orthogonal polynomials has various applications in mathematics, statistics, and other fields. It can be used to approximate functions, analyze data, and solve differential equations. It is also used in the study of probability and in the development of numerical methods for solving mathematical problems.

Are there any limitations or assumptions when using the limit of orthogonal polynomials?

Yes, there are limitations and assumptions when using the limit of orthogonal polynomials. One limitation is that it is only applicable to orthogonal polynomials, which have certain properties such as being mutually orthogonal. Additionally, the limit may not exist for certain types of polynomials or functions. Assumptions may also be made about the behavior of the polynomials as n increases, which may not always hold true in practical applications.

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