Bell multiplier sequences

Given a sequence of real numbers fkg1k=0, we can de ne an operator :R[x]!R[x] by [Pnk=0akxk] =Pnk=0kakxk. If has the property that it maps every hyperbolic polynomial to a hyperbolic polynomial, we callfkg1k=0a classical multiplier sequence. In place of the standard basis, we may consider a di erent polynomial basisQ=fqk(x)g1k=0to express a polynomialasPnk=0akqk(x) and [Pnk=0akqk(x)] =Pnk=0kakqk(x). If maps every hyperbolic polynomial to a hyperbolic polynomial with respect to the Q-basis,w e callfkg1k=0aQ-multiplier sequence. When the sequencefkg1k=0is of the formfp(k)g1k=0, wherep(x) is a polynomial with real coe cients, we say thatp(x) interpolates the sequencefkg1k=0, or that the sequencefkg1k=0ispolynomially interpolated.In this thesis, we study multiplier sequences for the Bell polynomial basis, referred to as Bell multiplier sequences. We begin with establishing thee xistence of trivial Bell multiplier sequences. We also disprove the existence of nontrivial linearly, geometrically, and quadratically interpolated Bell multiplier sequences. We conclude with a discussion of open questions inregards to Bell multiplier sequences.