Preliminary list of abstracts

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B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness
Keywords: Linear sampling algorithm; Quasi-interpolant representation; Besov space of mixed smoothness
Sun, 14:50--15:15
  • Dinh, Dũng (Information Technology Institute, Vietnam National University, Hanoi, Vietnam)

Let $\xi = \{x^j\}_{j=1}^n$ be a set of $n$ sample points in the $d$-cube $[0,1]^d$, and $\Phi = \{\varphi_j\}_{j =1}^n$ a family of $n$ functions on $[0,1]^d$. We define the linear sampling algorithm $L_n(\Phi,\xi,\cdot)$ for an approximate recovery of a continuous function $f$ on $[0,1]^d$ from the sampled values $f(x^1), ..., f(x^n)$, by \begin{equation*} L_n(\Phi,\xi,f) := \sum_{j=1}^n f(x^j)\varphi_j. \end{equation*} For the Besov class $B^\alpha_{p,\theta}$ of mixed smoothness $\alpha$, to study optimality of $L_n(\Phi,\xi,\cdot)$ in $L_q([0,1]^d)$ we use the quantity \begin{equation*} r_n(B^\alpha_{p,\theta})_q := \inf_{\xi, \Phi} \sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,\xi,f)\|_q, \end{equation*} where the infimum is taken over all sets of $n$ sample points $\xi = \{x^j\}_{j=1}^n$ and all families $\Phi = \{\varphi_j\}_{j=1}^n$ in $L_q([0,1]^d)$. We explicitly constructed linear sampling algorithms $L_n(\Phi,\xi,\cdot)$ on the set of sample points $\xi = G^d(m):= \{(2^{-k_1}s_1,...,2^{-k_d}s_d) \in [0,1]^d : k_1 + ... + k_d \le m\}$, with $\Phi$ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order $r$. For various $0<p,q,\theta \le \infty$ and $1/p < \alpha < r$, we proved upper bounds for the worst case error $ \sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,\xi,f)\|_q$ which coincide with the asymptotic order of $r_n(B^\alpha_{p,\theta})_q$ in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions $f \in B^\alpha_{p,\theta}$ by mixed B-spline series with the coefficient functionals which are explicitly constructed as linear combinations of an absolute constant number of values of functions. Moreover, we proved that the quasi-norm of the Besov space is equivalent to a discrete quasi-norm in terms of the coefficient functionals.