Abstract
Given gridded cell-average data of a smooth multivariate function, we present a constructive explicit procedure for generating a high-order global approximation of the function. One contribution is the derivation of high-order approximations to point-values of the function directly from the cell-average data. The second contribution is the development of univariate B-spline-based high-order quasi-interpolation operators using cell-average data. Multivariate spline quasi-interpolation approximation operators are obtained by tensor product of the univariate operators.
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Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Sergio Amat and Juan Ruiz-Alvarez have been supported through project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia) and by the national research project PID2019-108336GB-I00. Dionisio F. Yáñez has been supported by GVA project CIAICO/2021/227 and by grant PID2020-117211GB-I00 funded by MCIN/AEI/10.13039/501100011033.
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Appendix: A From cell-average to point values: 2D case
Appendix: A From cell-average to point values: 2D case
In this appendix, we suppose that \(m^{x},m^{y},p\in \mathbb {N}\) with 2mμ + 2 ≤ p, μ = x,y; \(x_{1},x_{2},y_{1},y_{2},h_{x},h_{y}\in \mathbb {R}\) with x1 < x2, y1 < y2, hx,hy > 0 and \(f\in \mathcal {C}^{p}([x_{1},x_{2}]\times [y_{1},y_{2}])\) with [a − mxhx,a + mxhx] × [b − myhy,b + myhy] ⊂ [x1,x2] × [y1,y2]. Also, we consider \(\tilde {\Omega }=[a-\frac {h_{x}}{2},a+\frac {h_{x}}{2}]\times [b-\frac {h_{y}}{2},b+\frac {h_{y}}{2}]\) and
Since \(f\in \mathcal {C}^{p}(\tilde {\Omega })\) then:
We define the function:
By Leibnitz rule, \(F\in \mathcal {C}^{p}([x_{1},x_{2}])\), then following the analysis in Section 2.1 we get:
We denote
noting that \(f_{a} \in \mathcal {C}^{p}([y_{1},y_{2}])\). Using the same formula for \(\bar {f}_{a}(b)\), we get:
Therefore, by (A.1) and (A.2), we have that:
For \(i\in \mathbb {N}\), we denote:
Subsequently, we know that for all y ∈ [y1,y2] we have by Section 2 that there exists ξ ∈ [x1, x2] such that:
where K is a constant independent on y; fy(x) = f(x,y) and
Note that \(\frac {\partial ^{2m^{x}+2}f}{\partial x^{2m^{x}+2}}(\xi ,y)\) is continuous in [y1, y2] since \(f\in \mathcal {C}^{p}([x_{1},x_{2}]\times [y_{1},y_{2}])\) (so \(\frac {\partial ^{2m^{x}+2}f}{\partial x^{2m^{x}+2}}(\xi ,y)\in {\mathscr{L}}^{1}([y_{1},y_{2}])\)). Thus,
Now, we can write that,
Collecting (A.3), (A.4) and (A.6), we get
with a0 = 1. We perform the following example in order to clarify the ideas proposed in this appendix.
Example A.1
For example, if mx = my = 1 and h = hx = hy, we know by Section 2 that a1 = − 1/24, then:
Again, the next result gives an exact formula for polynomials.
Corollary A.1
Let \(m^{x},m^{y}\in \mathbb {N}\) and \(P\in {\Pi }^{(2m^{x}+1,2m^{y}+1)}_{2}(\mathbb {R})\), then
being a0 = 1 and \(\mathbf {a}_{m}=(a_{i})_{i=1}^{m}\) the solution of the system (17).
Proof
In order to prove the corollary we introduce two considerations:
-
As \(P(x,y) \in {\Pi }^{2m^{x}+1,2m^{y}+1}_{2}(\mathbb {R})\), then the function
$$ F(x)=\frac{1}{h_{x}}{\int}_{b-\frac{h_{y}}{2}}^{b+\frac{{h_{y}}}{2}} P(x,y)dy, $$is a polynomial of degree 2mx + 1 on the variable x, thus
$$ \frac{\partial^{2m^{x}+2} F}{\partial x^{2m^{x}+2}}(x)=0, \forall x \in \mathbb{R}. $$(A.9) -
For all \(\mu \in \mathbb {R}\), the function defined as Pμ(y) = P(μ,y) is a polynomial of degree 2my + 1 on the variable y, thus
$$ \frac{\partial^{2m^{y}+2}P_{\mu}}{\partial y^{2m^{y}+2}}(y)=0, \forall y \in \mathbb{R}. $$(A.10)
With these considerations, the proof is similar to the construction of the formula (A.7) since:
By Section 2 and (A.9), we have that the formula for the 1d case is exact, i.e.,
being \(\mathbf {a}_{m^{x}}=(a_{i})_{i=1}^{m^{x}}\) the solution of the system in (17). Again, using the same formula for \(\bar {p}_{a}(b)\) and (A.10):
Thus, by (A.11) and (A.12), we have that:
By \(P(x,y)\in {\Pi }^{2m^{x}+1,2m^{y}+1}_{2}(\mathbb {R})\), we have for each \(y\in \mathbb {R}\)
Thus,
Then,
Finally, by (A.12), (A.16) and (A.15), we obtain:
with a0 = 1. □
Finally, we show a new example with different values mx and my.
Example A.2
For example, if mx = 2, my = 1 and h = hx = hy, we know by Section 2 that a1 = − 1/24,a2 = 3/640, then:
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Amat, S., Levin, D., Ruiz-Álvarez, J. et al. Explicit multivariate approximations from cell-average data. Adv Comput Math 48, 85 (2022). https://doi.org/10.1007/s10444-022-09997-5
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DOI: https://doi.org/10.1007/s10444-022-09997-5