Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials - MathOverflow

Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

2011-11-21T21:50:48Z

I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I need, but they are defined over the interval $[-1,1]$, and a change of variables ends up changing the weight function.

None of the orthogonal polynomial families I have looked at (Chebyshev, Gegenbauer, Legendre, Laguerre, Jacobi, Hermite) have this property.

Does anyone know of a family that does? Suggestions for references that may point the way would also be very helpful.

Thanks!

Answer by robot for Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

2011-11-21T23:09:57Z

If your polynomial is related to symmetries in one way or another it may be a matrix coefficient of a Lie group representation.

Answer by J. M. for Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

2011-11-22T04:32:23Z

There is this paper and this paper which treat the special case of "half-range Chebyshev polynomials" (both kinds, corresponding to the weights $\dfrac1{\sqrt{1-x^2}}$ and $\sqrt{1-x^2}$ over $[0,1]$) to deal with Fourier expansions of nonperiodic functions. I have a feeling that half-range Gegenbauer polynomials have been treated before, and I'll try to see what I can dig up.

In the meantime, one can use the Stieltjes procedure to build up the recursion relations for these half range Gegenbauers. Letting

$$\langle f(x),g(x) \rangle^{(\alpha)}=\int_0^1 (1-t^2)^{\alpha-1/2} f(t)g(t)\mathrm dt$$

be the associated inner product, the Stieltjes procedure for generating monic orthogonal polynomials $\phi_k(x)$ uses the formulae

$$\begin{align*}b_k&=\frac{\langle x\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}{\langle\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}\\ c_k&=\frac{\langle\phi_k(x),\phi_k(x)\rangle^{(\alpha)}}{\langle\phi_{k-1}(x),\phi_{k-1}(x)\rangle^{(\alpha)}}\end{align*}$$

to give the coefficients $b_k,c_k$ for the recursion relation

$$\phi_{k+1}(x)=(x-b_k)\phi_k(x)-c_k\phi_{k-1}(x)$$

Here, the result

$$\int_0^1 (1-t^2)^{\alpha-1/2}t^k \mathrm dt=\frac{\Gamma\left(\frac{1+k}{2}\right)\Gamma\left(\alpha+\frac12\right)}{2\Gamma\left(\alpha+\frac{k}{2}+1\right)}$$

is useful.

I might as well throw this in. There is an algorithm due to Chebyshev (1859) for determining recursion coefficients from the moments. I've already talked about the algorithm here, so I shall not repeat myself. Instead, I'll reproduce the Mathematica routine I gave in that answer:

chebAlgo[mom_?VectorQ, prec_: MachinePrecision] := 
 Module[{n = Quotient[Length[mom], 2], si = mom, ak, bk, np, sp, s, v},
  np = Precision[mom]; If[np === Infinity, np = prec];
  ak[1] = mom[[2]]/First[mom]; bk[1] = First[mom];
  sp = PadRight[{First[mom]}, 2 n - 1];
  Do[
   sp[[k - 1]] = si[[k - 1]];
   Do[
    v = sp[[j]];
    sp[[j]] = s = si[[j]];
    si[[j]] = si[[j + 1]] - ak[k - 1] s - bk[k - 1] v;
    , {j, k, 2 n - k + 1}];
   ak[k] = si[[k + 1]]/si[[k]] - sp[[k]]/sp[[k - 1]];
   bk[k] = si[[k]]/sp[[k - 1]];
   , {k, 2, n}];
  N[{Table[ak[k], {k, n}], Table[bk[k], {k, n}]}, np]
  ]

Here for instance is how to use chebAlgo[] to generate recursion coefficients for the monic half-range Chebyshev polynomials of the first kind:

With[{a = 0}, chebAlgo[Table[Gamma[(k + 1)/2] Gamma[a + 1/2]/Gamma[a + k/2 + 1],
                             {k, 0, 10}]/2, Infinity]] // FullSimplify

Answer by Ben Adcock for Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

2011-11-22T22:28:10Z

Actually, quite a lot is known about such polynomials, at least in the asymptotic regime $n \rightarrow \infty$, where $n$ is the index of the $n$th orthogonal polynomial. In the paper there are asymptotic formulae for not only the polynomials themselves, but also the coefficients of the recurrence relation. On could, in theory, use such formulae to compute recurrence coefficients (for large n), combined with a standard algorithm (such as the one posted by J. M.) for coefficients corresponding to small n.

Answer by JFvD for Orthogonal polynomials/functions on the interval [0,1] but with same weight as Gegenbauer polynomials

2013-01-01T19:30:29Z

The even (degree) Gegenbauer polynomials $C_{2n}^{(\alpha )}$, $n=0,1,2,\ldots$ form an orthogonal basis for the space of square integrable functions over [0,1] with respect to the weight function $(1-x^2)^{\alpha -1/2}$, and so do the odd (degree) Gegenbauer polynomials $C_{2n+1}^{(\alpha )}$, $n=0,1,2,\ldots$