On the best Ulam constant of a higher order linear difference equation

2020/07/09 by Baias, Alina Ramona, Popa, Dorian
#39A30 #39B62 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2007.04654

Abstract

In a Banach space X the linear difference equation with constant coefficients x_n+p = a₁x_n+p-1 +… + a_px_n, is Ulam stable if and only if the roots r_k, 1≤ k≤ p, of its characteristic equation do not belong to the unit circle. If |r_k| > 1, 1≤ k≤ p, we prove that the best Ulam constant of this equation is (1)/(|V|)∑_s=1^∞|(V₁)/(r₁^s)-(V₂)/(r₂^s)+… +\frac(-1)^p+1V_pr_p^s|, where V = V (r₁, r₂, …, r_p) and V_k =V (r₁,…, r_k-1, r_k+1,…, r_p), 1≤ k≤ p, are Vadermonde determinants.

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