A family of non-oscillatory 6-point interpolatory subdivision schemes

Abstract

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Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

“…For linear stationary subdivision schemes, Lipschitz stability is a consequence of convergence, but this is not the case for nonlinear subdivision [1,5,16]. Some theory was developed and successfully applied on several instances [1,3,5,7,9,17].…”

“…However, as in section 7, we shall be able to prove stability for a conveniently restricted class of strictly monotone data. For such data, the order of approximation can be obtained by the usual, Taylor-like, one-step approximation results [9]. 8.1.…”

“…Stability. In [1,3,9], stability is proved using a result similar to Theorem 3 (see for instance Theorem 1 of [9]), which requires that the scheme is of the form (10).…”

High-accuracy approximation of piecewise smooth functions using the Truncation and Encode approach

“…A subdivision scheme is convergent if this process converges to a continuous function. See for instance [4,9,10].…”

“…Figure 1 shows a sketch of recurrence (9). It has a simple interpretation: On one hand, if |d k−1 j | < ε k , then a 'small' prediction error indicates that P k k−1 approximates well the function at ξ k 2 j+1 ; thus P k+1 k may also provide a good approximation at ξ k+1 2i+1 , i ∈ {2 j, 2 j + 1}, and we do not need to evaluate f at these points.…”