Oppolynomials

Oppolynomials are a hypothetical class of polynomials studied in the context of orthogonality and duality in approximation theory. They are defined relative to a base weight function w on a finite interval [a,b] and a corresponding dual weight obtained by reflecting the interval about its midpoint, namely w_opp(x) = w(a + b − x).

A sequence {O_n(x)} of polynomials with degree n is called the oppolynomial sequence for w if O_0(x) = 1, O_1(x) = x, and they satisfy a three-term recurrence of the form O_{n+1}(x) = (A_n x + B_n) O_n(x) − C_n O_{n−1}(x). The numbers A_n, B_n, C_n are chosen so that the polynomials are orthogonal with respect to the dual inner product defined by w_opp: ∫_a^b O_n(x) O_m(x) w_opp(x) dx = h_n δ_{nm}, where δ_{nm} is the Kronecker delta.

Properties of oppolynomials mirror many features of classical orthogonal polynomials. When w_opp is positive on [a,b], a Gram–Schmidt construction yields a unique monic sequence, and the zeros of O_n are real and lie in [a,b] for n ≥ 1. There is also a duality relation with standard orthogonal polynomials P_n associated with w; in some cases O_n(x) is related to P_n(a+b−x) up to a normalization factor, particularly when w is symmetric about the midpoint.

Computation can proceed via Gram–Schmidt using the inner product with w_opp or via the three-term recurrence once the moments of w_opp are known. If w is symmetric, w_opp = w and oppolynomials coincide with the usual orthogonal polynomials for w.

Applications of the concept appear in duality studies of polynomial families and in quadrature contexts involving mirrored domains. See also orthogonal polynomials, dual polynomials, and Gaussian quadrature.