Eventual Scalar Recurrences for Constrained Orthogonal Polynomials

Abstract

Let S be a finite-codimensional subspace of R[x] with a positive definite moment inner product, and let q_n be its degree-ordered monic orthogonal basis. For a fixed nonzero polynomial a of degree d, we prove that an eventual scalar band recurrence for a q_n exists if and only if multiplication by a preserves S plus its polynomial orthogonal complement. The width can then be chosen to be d. Some nonconstant multiplier works exactly when this sum contains a nonzero polynomial ideal. For the inner product on [0,1] and the single constraint integral_2^3 p(x) dx = 0, every nonconstant multiplier instead sends infinitely many q_n outside S. A mean-zero Legendre example shows why stabilizing S itself is not necessary for a tail recurrence. These elementary results refine an established stabilizer approach; they do not classify differential, coupled, or compressed-operator recurrences, or settle the broader AIM problem. The note is self-audited and unrefereed; absolute priority and independent review are not claimed.