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7, there is no assumption that the ri , i = k are constant with respect to the limit process. 15) shows that lim ||Dk − rk →∞ k ||∞ = lim ||Dk+1 − rk →∞ k ||∞ = 0. 3. 12) with i = k. The following proposition shows that the influence of rk in this case has an exponential decay away from the interval tk − tk+1 . 9. (Exponential decay property) Let Di , i = 1, . . , n − 1, denote the solution of the consistency equations with tension parameters rk ≥ r > 2, i = 0, . . , n − 1, and let Dˆ i , i = 1, .

On [t0 , tn ], lim || p − l|| = 0. 17) r →∞ Proof. Suppose ri = r, i = 0, . . 15), it follows that (h i i−1 + h i−1 i ) , i = 1, . . , n − 1. 19) where || || = max || 1≤i≤n−1 i ||∞ . 20) Hence the solution Di , i = 1, . . 15) is bounded with respect to r . 4 can clearly be extended to the case of bounded r . 17). 7. (Local tension property) Let ri ≥ r > 2, ∀i and consider an interval tk , tk+1 for a fixed k ∈ {0, . . , n − 1}. e. lim rk →∞ pk − lk = 0. 21) Proof. 6 (where we can assume the additional constraints rk ≥ r > 2 to the hypotheses currently being imposed.

Suppose ri = r, i = 0, . . 15), it follows that (h i i−1 + h i−1 i ) , i = 1, . . , n − 1. 19) where || || = max || 1≤i≤n−1 i ||∞ . 20) Hence the solution Di , i = 1, . . 15) is bounded with respect to r . 4 can clearly be extended to the case of bounded r . 17). 7. (Local tension property) Let ri ≥ r > 2, ∀i and consider an interval tk , tk+1 for a fixed k ∈ {0, . . , n − 1}. e. lim rk →∞ pk − lk = 0. 21) Proof. 6 (where we can assume the additional constraints rk ≥ r > 2 to the hypotheses currently being imposed.