Explicit orthogonal bases for generalized Jacobi weights on connected interval families

Obtain explicit orthogonal bases for interval families with a connected endpoint graph using the moment reduction formulas for generalized Jacobi weights.

Background

For generalized Jacobi weights, the paper reduces cell moments to finite linear combinations of classical Jacobi cell moments and gives endpoint representations for positive-degree moments. Separately, connected endpoint graphs guarantee that the associated moment form is an inner product and therefore yield an orthogonal polynomial basis abstractly. The unresolved problem is to combine these results to produce explicit orthogonal bases for such connected interval families.

References

Several questions remain open. It would be interesting to derive explicit formulas or recurrence relations for the orthogonal basis associated with the endpoint graph and to determine more precisely how the topology of the corresponding tree affects the diagonal entries and the spectral condition number of the moment matrix. Finally, it would be interesting to use the moment reduction formulas for generalized Jacobi weights to obtain explicit orthogonal bases for interval families with a connected endpoint graph.

— Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation  (2608.25714 - Guessab et al., 26 Aug 2026) in Section 6, Conclusions and Future Work