12 MICHAEL FALK, MAXIMILIAN ZOTT [0 , 1] $^{k}$. Put for i = 1 , . . . , d and the bandwidth h > 0 Define for i = 1 , . . . , d which is the set of those points t ∈ [0 , 1] k that are closest to the grid point s$_{i}$ . Lemma 3.1. We have for arbitrary t ∈ [0 , 1] k and 1 ≤ i ≤ d as well as g$_{i,h}$ ( t ) ≤ 1 . Proof. The convergence g$_{i,h}$ ( s$_{i}$ ) →$_{h}$$_{↓}$$_{0}$ 1 follows from the fact that K (0) = 1 and that the D -norm of a unit vector is 1. The fact that an arbitrary D -norm is bounded below by the sup-norm together with the monotonicity of K implies for t ∈ [0 , 1] k Note that K ( ‖ t − s$_{i}$ ‖ /h ) /K (min$_{1}$$_{≤}$$_{j}$$_{≤}$$_{d}$ ‖ t − s$_{j}$ ‖ /h ) →$_{h}$$_{↓}$$_{0}$ 0 if t ̸∈ N ( s$_{i}$ ) by the required growth condition on the kernel K in (6). □ The above Lemma shows in particular g$_{i,h}$ ( s$_{j}$ ) →$_{h}$$_{↓}$$_{0}$ δ$_{ij}$ which is close to condi- tion (5). Obviously, the functions g$_{i,h}$ are constructed in such a way that condition (2) holds exactly. Therefore, we obtain the generalized max-linear model which does not interpolate ( η$_{s}$$_{1}$, . . . , η$_{s}$$_{d}$ ) exactly, but ˆ η$_{s}$$_{i}$$_{,h}$ converges to η$_{s}$$_{i}$ as h ↓ 0. Note that the limit functions lim$_{h}$$_{↓}$$_{0}$ g$_{i,h}$ are not necessarily continuous: For instance, there may be t$_{0}$ ∈ [0 , 1] k with ‖ t$_{0}$ − s$_{1}$ ‖ = · · · = ‖ t$_{0}$ − s$_{d}$ ‖ . Then t$_{0}$ ∈ g$_{i,h}$ ( t ) := K ( ‖ t − s$_{i}$ ‖ /h ) ‖ ( K ( ‖ t − s$_{1}$ ‖ /h ) , . . . , K ( ‖ t − s$_{d}$ ‖ /h )) ‖$_{D}$$_{s}$$_{1}$$_{,...,s}$$_{d}$ , t ∈ [0 , 1] $^{k}$. (7) N ( s$_{i}$ ) := { t ∈ [0 , 1] k : ‖ t − s$_{i}$ ‖ ≤ ‖ t − s$_{j}$ ‖ , j = i } , g$_{i,h}$ ( t ) →$_{h}$$_{↓}$$_{0}$    $^{1}$ , if t = s$_{i}$ 0 , if t ̸∈ N ( s$_{i}$ ) g$_{i,h}$ ( t ) ≤ K ( ‖ t − s$_{i}$ ‖ /h ) max$_{1}$$_{≤}$$_{j}$$_{≤}$$_{d}$ K ( ‖ t − s$_{j}$ ‖ /h ) = K ( ‖ t − s$_{i}$ ‖ h ) K ( min$_{1}$$_{≤}$$_{j}$$_{≤}$$_{d}$ ‖ t − s$_{j}$ ‖ h ) ≤ 1 . ˆ η$_{t,h}$ = max i =1 ,...,d η$_{s}$$_{i}$ g$_{i,h}$ ( t ) , t ∈ [0 , 1] $^{k}$,