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.