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

Let F k be a mapping from R Z to R Z , satisfying that for x ∈ R Z and n ∈ Z , F k ( x ) ( n ) is the ( k + 1 ) th largest value (median value) of the 2 k + 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 n 0 ∈ Z such that x is locally finitely convergent with respect to F k on { n 0 , … , n 0 + k − 1 } , then x is finitely convergent with respect to F k . 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.