Stability of linear predictors and numerical range of shift operators in normed spaces

Abstract — The zeros of predictor polynomials are shown to belong to the numerical range of a shift operator associated with the particular prediction problem under consideration. The numerical range consists of the classical field of values of the shift operator when the setting is Hilbert space, but a new definition is necessary when the setting is a general normed space. It is shown that a predictor polynomial is not stable in general. Nevertheless, for predictor polynomials in ℓp spaces, it is shown that their zeros belong to the open circular disk with radius 2. Index Terms — Linear prediction, stability, numerical range, numerical radius, ℓp norms.