Norms of polynomials of the Volterra operator

Volterra operator norms : a brief survey

“…Figure 6 suggests that there is a region of pairs (ξ, η) around (−1, 1) where ∥I + ξV + ηV 2 ∥ = 1. The following result established in [17], proves this rigorously.…”

“…To our knowledge, the only case that has been investigated in detail, beyond the linear case, is when p is a quadratic polynomial with real coefficients. This was done recently in [17], which describes both the theoretical background and the numerical implementation. Figure 5 is a contour plot of the function (σ, τ ) → ∥V 2 + σV + τ I∥, computed using the results in [17].…”

“…This was done recently in [17], which describes both the theoretical background and the numerical implementation. Figure 5 is a contour plot of the function (σ, τ ) → ∥V 2 + σV + τ I∥, computed using the results in [17].…”

“…A heuristic argument suggests trying real-quadratic polynomials. Indeed, if there is a polynomial p such that ∥p(V )∥/ max W (V ) |p| > 2, then, since ∥V n ∥ → 0 very rapidly with n, one might expect lower powers of V to dominate in p(V ), and since W (V ) is symmetric about the x-axis, one might expect the same to be true of the roots of p. The authors of [17] computed the ratio ∥p(V )∥/ max W (V ) |p| for real-quadratic polynomials. Figure 9 illustrates the results of these computations.…”

Entropy-metric estimation of the small data models with stochastic parameters

“…We present the discrete dynamic small data model for the above interpretation of the studied process based on the Volterra polynomial [ 30 , 31 ] as …”