A note on properties of locally finitely convergent sequences with respect to median filter

Let Fk be a mapping from RZ to RZ, satisfying that for x∈RZ and n∈Z, Fk(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775-2781], we conjectured that for k∈{2,3}, if there exists n0∈Z such that x is locally finitely convergent with respect to Fk on {n0,…,n0+k−1}, then x is finitely convergent with respect to Fk. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true.