Quasi-interpolation in Spline Spaces: Local Stability and Approximation Properties

In this work we analyze the approximation error in Sobolev norms for quasi-interpolation operators in spline spaces. We establish in a general way the hypotheses on a quasi-interpolant to achieve the optimal order of approximation. Finally, we propose simple but general constructions of such operators that satisfy the established hypotheses and illustrate their performance through some numerical tests.

Keywords:
spline approximation; quasi-interpolation; stability; B-splines

Authorship

M. E. CASTILLO

Departamento de Matemática, FACET, Universidad Nacional de Tucumán, Tucumán, Argentina - E-mail: mariaemiliacastillo@gmail.comUniversidad Nacional de TucumánArgentinaTucumán, ArgentinaDepartamento de Matemática, FACET, Universidad Nacional de Tucumán, Tucumán, Argentina - E-mail: mariaemiliacastillo@gmail.com

https://orcid.org/0000-0003-1702-4271

E. M. GARAU ^**Corresponding author: Eduardo M. Garau - E-mail: eduardogarau@gmail.com

Universidad Nacional del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas, FIQ, Santa Fe, Argentina - E-mail: eduardogarau@gmail.comUniversidad Nacional del LitoralArgentinaSanta Fe, ArgentinaUniversidad Nacional del Litoral, Consejo Nacional de Investigaciones Científicas y Técnicas, FIQ, Santa Fe, Argentina - E-mail: eduardogarau@gmail.com

https://orcid.org/0000-0001-7541-2390

*Corresponding author: Eduardo M. Garau - E-mail: eduardogarau@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (5)
Formulas (19)

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Figure 1:
Examples of cubic B-splines of maximum smoothness (top) and with a triple internal knot (bottom).

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Figure 2:
The procedure from (i) for the choice of coefficients c _I,β in (4.3) for some B-splines β (solid lines).

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Figure 3:
L²-error in different approximation methods for f in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.

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Figure 4:
L²-error in different approximation methods for g in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.

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Figure 5:
The decay of

‖ g - Q_{1} (g) ‖_{L^{2}}

for splines of degrees 4 (left) and 5 (right) with maximum smoothness and globally continuous splines of the same degrees onto uniform and adaptive meshes.

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(4.1)

(4.2)

(4.3)

Figure 1: Examples of cubic B-splines of maximum smoothness (top) and with a triple internal knot (bottom).

Figure 2: The procedure from (i) for the choice of coefficients c _I,β in (4.3) for some B-splines β (solid lines).

Figure 3: L²-error in different approximation methods for f in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.

Figure 4: L²-error in different approximation methods for g in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.

Figure 5: The decay of $‖ g - Q_{1} (g) ‖_{L^{2}}$ for splines of degrees 4 (left) and 5 (right) with maximum smoothness and globally continuous splines of the same degrees onto uniform and adaptive meshes.

Ξ = {\underset{m_{1} times}{\underset{⏟}{ζ_{1}, \dots, ζ_{1},}} \underset{m_{2} times}{\underset{⏟}{ζ_{2}, \dots, ζ_{2},}} \dots \underset{m_{N} times}{\underset{⏟}{ζ_{N}, \dots, ζ_{N}}}} .

W^{r, q} (E) : = {u : E \to ℝ ∣ u, D u, \dots, D^{r - 1} u are absolutely continuous on E and D^{r} u \in L^{q} (E)} .

(3.1)

Q f : = \sum_{β \in B} λ_{β} (f) β,

| λ_{β} (f) | \leq C_{Q} | \ supp β |^{- \frac{1}{q}} ‖ f ‖_{L^{q} (supp β)},

(3.2)

‖ Q f ‖_{L^{q} (I)} \leq C_{Q} ‖ f ‖_{L^{q} (\tilde{I})}, \forall f \in L^{q} (\tilde{I}) .

(3.3)

‖ f - Q f ‖_{L^{q} (I)} \leq \frac{(1 + C_{Q})}{ℓ!} | \tilde{I} |^{ℓ + 1} ‖ D^{ℓ + 1} f ‖_{L^{q} (\tilde{I})}, \forall f \in W^{ℓ + 1, q} (\tilde{I}),

‖ f - Q f ‖_{L^{q} (I)} \leq ‖ f - g ‖_{L^{q} (I)} + ‖ Q (g - f) ‖_{L^{q} (I)} \leq (1 + C_{Q}) ‖ f - g ‖_{L^{q} (\tilde{I})} .

‖ f - Q f ‖_{L^{q} [a, b]} \leq C | I |^{ℓ + 1} ‖ D^{ℓ + 1} f ‖_{L^{q} [a, b]}, \forall f \in W^{ℓ + 1, q} ([a, b]),

‖ D^{r} (Q f) ‖_{L^{q} (I)} \leq C_{p, r} (p + 1) C_{Q} | I |^{- r} ‖ f ‖_{L^{q} (\tilde{I})}, \forall f \in L^{q} (\tilde{I}),

D^{r} (Q f) (x) | = |\sum_{i = k - p}^{k} λ_{i} (f) D^{r} β_{i} (x)| \leq \max_{k - p \leq i \leq k} | D^{r} β_{i} (x) | \sum_{i = k - p}^{k} | λ_{i} (f) | \leq C_{p, r} | I |^{- r} \sum_{i = k - p}^{k} | λ_{i} (f) | .

θ^{- 1} \leq \frac{ζ_{j} - ζ_{j - 1}}{ζ_{j + 1} - ζ_{j}} \leq θ, \forall j = 2, \dots, N - 1 .

(3.4)

‖ D^{r} (Q f) ‖_{L^{q} (I)} \leq C_{S} ‖ D^{r} f ‖_{L^{q} (\tilde{I})}, \forall f \in W^{r, q} (\tilde{I}),

‖ D^{r} (Q f) ‖_{L^{q} (I)} \leq ‖ D^{r} Q (f - g) ‖_{L^{q} (I)} \leq \frac{C_{p, r} (p + 1) C_{Q}}{| I |^{r}} ‖ f - g ‖_{L^{q} (\tilde{I})} \leq \frac{C_{p, r} (p + 1) C_{Q}}{(r - 1)!} \frac{| \tilde{I} |^{r}}{| I |^{r}} ‖ D^{r} f ‖_{L^{q} (\tilde{I})} .

(3.5)

‖ D^{r} (f - Q f) ‖_{L^{q} (I)} \leq \frac{(1 + C_{S})}{(ℓ - r)!} | \tilde{I} |^{ℓ + 1 - r} ‖ D^{ℓ + 1} f ‖_{L^{q} (\tilde{I})}, \forall f \in W^{ℓ + 1, q} (\tilde{I}),

‖ D^{r} (f - Q f) ‖_{L^{q} (I)} \leq ‖ D^{r} (f - g) ‖_{L^{q} (I)} + ‖ D^{r} Q (g - f) ‖_{L^{q} (I)} \leq (1 + C_{S}) ‖ D^{r} (f - g) ‖_{L^{q} (\tilde{I})} .

(4.1)

Q f : = \sum_{β \in B} λ_{β} (f) β,

Π_{I} f = \sum_{i = 1}^{p + 1} λ_{i}^{I} (f) β_{i}^{I}, \forall f \in L^{1} (I),

(4.2)

\max_{i = 1, \dots, p + 1} | λ_{i}^{I} (f) | \leq C | I |^{- \frac{1}{q}} ‖ f ‖_{L^{q} (I)}, f \in L^{q} (I),

(4.3)

λ_{β} : = \sum_{I \in I_{β}} c_{I, β} λ_{β}^{I},

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Quasi-interpolation in Spline Spaces: Local Stability and Approximation Properties

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[1] *Corresponding author: Eduardo M. Garau - E-mail: eduardogarau@gmail.com