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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!

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  • $\begingroup$ The orthogonal polynomials for this weight function probably won't be anything familiar: already for $\alpha=0$ and degree $1$ the polynomial is a multiple of $\pi x - 2$ ... $\endgroup$
    – Noam D. Elkies
    Commented Nov 21, 2011 at 22:12
  • $\begingroup$ Familiarity is perhaps not so important. What I really want in the end is to be able to derive triple-integral identities such as the ones for spherical harmonics (so that I can represent the product of two such basis elements as a sum over this basis). $\endgroup$
    – Marcus P S
    Commented Nov 21, 2011 at 22:22
  • $\begingroup$ That should be "triple-product integral identities". $\endgroup$
    – Marcus P S
    Commented Nov 22, 2011 at 1:10

4 Answers 4

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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
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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.

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  • $\begingroup$ The low order coefficients and polynomials are actually the most relevant for the application I have in mind. The reference you pointed to seems to focus on orthogonal polynomials over [-1,1] --- or am I missing something? $\endgroup$
    – Marcus P S
    Commented Nov 24, 2011 at 15:14
  • $\begingroup$ You can map $[0,1]$ to $[-1,1]$ in the obvious way to use the results of the paper. What kind of application are you looking at? I'd be interested to see instances where these polynomials naturally arise (I've also done some work in the past with the "half-range Chebyshev polynomials" mentioned above). $\endgroup$
    – Ben Adcock
    Commented Nov 24, 2011 at 17:08
  • $\begingroup$ I am interested in performing Bayesian estimation of a quantum state. A natural way to parameterize pure quantum states is by using angles in a way very similar to spherical coordinates -- see for example eqn 4.69 of "Geometry of quantum states" by Ingemar Bengtsson and Karol Życzkowski. The resulting metric leads one to consider the problem above. There are known ways to do Bayesian estimation using special functions/polynomials families that are related to group representations (as r0b0t suggested below), but I was looking for a different approach. $\endgroup$
    – Marcus P S
    Commented Dec 30, 2011 at 5:57
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If your polynomial is related to symmetries in one way or another it may be a matrix coefficient of a Lie group representation.

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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 $C_{2n+1}^{(\alpha )}$, $n=0,1,2,\ldots$

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  • $\begingroup$ Notice that the classical orthogonal families that you tried have weight functions that are singular or vanishing at the endpoints of the interval whereas your weight function does not at the endpoint $x=0$. This explains why such strategies of transforming the interval fail. However, since the weight function in question is even, the Hilbert space over the interval [0,1] is isomorphic to the even and odd subspaces of the Hilbert space over the interval $[-1,1]$, so you can get an orthogonal basis through projection as indicated above. $\endgroup$
    – JFvD
    Commented Jan 3, 2013 at 13:24

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