Explicit characterization of endpoint-graph orthogonal bases and tree-dependent conditioning

Derive explicit formulas or recurrence relations for the orthogonal basis associated with the endpoint graph, and determine more precisely how the topology of the corresponding tree affects the diagonal entries and spectral condition number of the moment matrix.

Background

The paper constructs a weighted inner product on the polynomial space from interval moments whenever the endpoint graph of the interval family is connected. Applying Gram–Schmidt yields a unique monic orthogonal polynomial basis and a diagonal weighted Gram matrix, but the construction is implicit. The authors leave open the derivation of explicit formulas or recurrence relations for this basis and a more precise analysis of how the topology of the associated endpoint tree influences the resulting Gram-matrix diagonal entries and spectral condition number.

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.

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

But the question remains about defining $u$ at the edge of our interval $x_{N}$.

— Inverse eigenvalue problem for discrete three-diagonal Schroedinger operator and the continuum limit in Tikhonov regularization approach  (2610.18734 - Chabanov, 13 Oct 2026) in Section 2, discussion immediately preceding Eq. (vfdN)