Harmonic analysis of little \(q\)-Legendre polynomials

Abstract

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as \(L^1\)-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual behavior strongly depends on these underlying polynomials. We study the little \(q\)-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their \(L^1\)-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the \(L^1\)-algebras associated with the little \(q\)-Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these \(L^1\)-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any \(L^1\)-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior.