Legendre and Chebyshev polynomials appear very frequently in applications.

Legendre and Chebyshev polynomials are known orthogonal polynomials in [-1,1]. However, due to their least(minimum) error property the Chebyshev polynomials are frequently used in applications.

Certainly Chebyshev polynomials for many reasons, but especially for their better goodness of fit.

From my experience, Chebyshev polynomials have good numerical properities, and they are mostly used by people in Mathematics, for example, for the convergence analysis, or error bound estimates. Legendre polynomials are more liked by people in Physics, or mathematical physics, or marine engineering, etc, since Legendre polynomial and its extensions can be used to express the solutions of some PDEs.

I also would recommend the Legendre polynomials. As a student I used to learned about their applications in PDEs from the book of Bernardi, Maday, Rapetti.

orthogonality is defined w.r.t. a scalar product, and if the questioner meant

=\int_{-1}^1 f(x)g(x)dx,

then theanswer can be ''Legendre'' only, since the result is unique up to scaling.

Well, all answers right insofar indeed Chebyshev (first kind) is quite often used

because of their good properties in approximation. But these are orthogonal

w.r.t. weight 1/sqrt(1-x^2)

Yes, that is the benefit of Legendre polynomials, they are orthogonal without any weight function like other polynomials (Hermite, Laguerre).

This is hard to quantify, but in my opinion, the Chebychev polynomials would lead the pack.

1. 1Th chebychev Polynomial

2. Legendre Polynomial

3. B splines(but they should be orthogonalized by gram Schmidt procedure and fit to [-1,1])

Dear all,

Thank you for sharing your thoughts.

Dear all, The concept of orthogonal polynomials is very important and has many applications in applied Mathematics and other fields of science. The Chebyshev polynomials are fundamental in characterizing the solution of best approximation with different norms.

Dear Abedallah M Rababah,

The classical orthogonal polynomials are the most widely used orthogonal polynomials.

@Ismat, spreading sequences have nothing to do with orthogonal polynomials. The orthogonality used in CDMA is approximate. Classical OP are  orthogonal wrt an integral dot product like \int f(x)g(x) \mu(dx).

dears @Ismat and @Patrick, thank you for the contributions; it is highly appreciated.

Some interesting applications of Chebyshev  polynomials, the first two exploiting the Chebyshev minimax property.  The third makes use of the property that Tk(cos(x)) = cos(kx), where Tk(x) is the kth degree Chebyshev polynomial.

Dear Professor @Scott, thank you for the 3 important applications of the Chebyshev polynomials.

can anyone tell me in what respect legendre polynomial is better than chebeyshev , while using for polynomial approximation?

Dear @Vaishali, in almost all computational cases it is easier to solve a problem using the Legendre polynomials than the Chebyshev polynomials because the Legendre polynomial has the weight function w(x)=1; for example in the least-squares and for the Gaussian Quadratures.

Abedalla, in many cases, you're correct about the convenience of Legendre polynomials, especially for least-square type calculations.  I used Legendre polynomials (and their integrals) for some of my dissertation work.  However, recurrence relation T(n+1)=2xT(n) - T(n-1) and the trigonometric T(n)(cos(x)) = cos(nx) have helped derive fast and convenient computations with Chebyshev polynomials, too.

Yes indeed dear @Scott, the Chebyshev polynomials are as well important as the Legendre polynomials and using the roots of the Chebyshev polynomials as nodes of interpolation minimizes the error of the Lagrange interpolation. 

Dear Scott

There is a small typo (missing factor) in your recurrence relation.

It should read Tn+1(x) = 2xTn(x)  -  Tn-1(x)

Dear Abedallah

" ... using the roots of the Chebyshev polynomials as nodes of interpolation minimizes the error ...".  You probably mean "reduces" instead of "minimizes".

Interpolating over Chebyshev's nodes does not minimize the error bound. The error is the product of two terms but only one of which will be minimized. Thus, although the error bound will be reduced, the output generally will not coincide with the minimax solution (nor with a quasi-minimax solution which would be obtained via a telescoping procedure).

corrected, thank you.

Thanks dear @H.E. Lehtihet, yes indeed, using the roots of the Chebyshev polynomials as nodes of interpolation reduces the maximum error of the minimax approximation.

Yes the Legendre polynomial would be ideal. Do you need to use a polynomial? Could you also use complex sinusoids (i.e. frequency analysis)? 

Thank you dear Professor @Kennedy, I asked a classical question in Approximation Theory; many Mathematicians think that an orthogonal polynomial set is more important than others, but at the end each orthogonal polynomials set was created to serve for a proper purpose.

The Lagrange polynomials are also popular, which are based on Legendre or Chebyshev polynomials.