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The definition of infinite refinement in Royset & Wets (Multivariate epi-splines and evolving function identification problems, Set-Valued and Variational Analysis, 2016) needs to be strengthened to exclude refinements, as illustrated by an example, that would not become fine enough. Without this strengthening, some results in that paper will not hold. Date: October 12, 2016 The definition of an infinite refinement needs to be adjusted as follows: 3.4 Definition (infinite refinement) A sequence {R}ν=1 of partitions of a closed set B ⊆ IR, with Rν = {Rν k} ν k=1, is an infinite refinement if for every x ∈ B and ε > 0, there exist ν̄ ∈ IN and δ ∈ (0, ε) such that R k ⊂ IB(x, ε) for every ν ≥ ν̄ and k satisfying R k ∩ IB(x, δ) ̸= ∅. The reason is that the corresponding definition in [1] allows for the possibility to choose a somewhat pathological refinement that would fail to become sufficiently “small” as illustrated next: Example. Let B = [0,∞) ⊂ IR and the infinite refinement be partitioning (0, √ ν) mostly into segments of length 1/ √ ν, with the exception that a portion of the segment nearest the origin is added to the final segment covering ( √ ν,∞). Specifically, we let Nν = ν + 1, Rν 1 = (0, 1/(4 √ ν))∪ (1/(2 √ ν), 1/ √ ν), Rν k = ((k−1)/ √ ν, k/ √ ν) for k = 2, . . . , Nν −1, and Rν Nν = ( √ ν,∞)∪ (1/(4 √ ν), 1/(2 √ ν)). Clearly, for every ν ∈ IN , {Rν k} ν k=1 is a partition of B. Also, {R}ν=1 would be an appropriate infinite refinement of B according to [1], which requires only: for every x ∈ B and ε > 0, there exists ν̄ ∈ IN such that R k ⊂ IB(x, ε) for every ν ≥ ν̄ and k satisfying x ∈ clR k. We establish that this requirement is satisfied: Let x ∈ B and ε > 0 be arbitrary. First, suppose that x = 0. Then, set ν̄ ≥ 1/ε2. In this case, we have that x ∈ clRν 1 for all ν. Since Rν 1 ⊂ [0, 1/ √ ν] ⊂ [0, ε]

Erratum to: Multivariate Epi-splines and Evolving Function Identification Problems