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Korovkin-type convergence results for multivariate Shepard formulae. (English)
Rev. Anal. Numér. Théor. Approx. 38, No. 2, 170-176 (2009).
Let $X$ be a compact subset of $\Bbb R^d (d\geq 1)$ and $B_n=\{x_1,x_2,\dots, x_{c(n)}\}$ be an interpolation node distribution. For $α>1$, let $w_α(x)= \Vert x\Vert _2^{-α}$ and $$φ_α^{(i)}(x)=\frac {w_α(x-x_i)}{\sum_{j=1}^{c(n)} w_α(x-x_j)}, i\in\{1,2,\dots, c(n)\}, φ_α^{(i)}\in C^\infty(X,\Bbb R).$$ The operator $S_{B_n}^α:C(X,\Bbb R)\to C^\infty (X)$, defined by $$S_{B_n}^αf(x)=\sum_{i=1}^{c(n)} f(x_i)φ_α^{(i)}(x), x\in X,$$ is called the multivariate Shepard interpolant to $f$ (corresponding to $B_n$). Let $n_0\in\Bbb N$ and $B_{n_0}=\{x_1,x_2,\dots, x_{c(n_0)}\}\subset X\subset\Bbb R^d$ be an interpolation node ditribution. Then, the sequence $(B_n)_{n\geq n_0}$ is said to be monotone in $X$, provided there exists $η:\Bbb N\to\Bbb R$ such that $\lim_{n\to\infty} η(n)=0$ and (a) $B_n\subset B_{n+1}\subset X$ for all $n\geq n_0$; (b) $c(n)[r(B_n)]^{d+1}\leq η(n)$ for $n\geq n_0$, where $r(B_n)=\inf \{δ>0: \forall x\in X, {\text{ card }}(B_δ(x)\cap B_n)\geq 1\}$ holds true with $B_δ(x)=\{y\in X:\Vert x-y\Vert _2<δ\}$. Using this definition of monotony for sequences of interpolation node distributions and the convergence of $S_{B_n}^α$ for $2d+1$ monomial functions $e_i^j:x\to x(i)^j$, $1\leq i\leq d$, $0\leq j\leq 2$, the author proves the following: For $f\in C(X,\Bbb R)$, $S_{B_n}^αf$ denotes the appropriate Shepard interpolant corresponding to a monotone sequence of interpolation node distribution $(B_n)_{n\geq n_0}$. Then $$\lim_{n\to\infty}\Vert S_{B_n}^αf-f\Vert _\infty=0{\text{ and }} \vert S_{B_n}^αf(x)-f(x)\vert\leq 2ω(f,γ_n(x)), x\in X$$ $${\text{ when }}γ_n^2 (x)=S_{B_n}^α\left( \sum_{i=1}^d(\cdot-x(i)^2\right)(x).$$
Reviewer: Costica Mustata (Cluj-Napoca)