Related papers: Orthogonal projectors onto spaces of periodic spli…

The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{\Delta_n}(f))$ onto splines of order $k$ with arbitrary knots $\Delta_n$, converges almost everywhere provided that the…

Let $d\in\mathbb N$ and $f$ be a function in the Orlicz class $L(\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\mathbb{R}^d$. Given partitions $\Delta_1,\ldots,$ $\Delta_d$ of $[0,1]$, we first prove that the orthogonal projection…

Let $t_{i}=\frac{i}{n}$ for $i=0,...,n$ be equally spaces knots in the unit interval $[0,1].$ Let $\mathcal{S}_{n}$ be the space of piecewise linear continuous functions on $[0,1]$ with knots $\pi_{n}=\{t_{i}:0\leq i\leq n\}.$ Then we have…

We prove that the Chebyshev spline orthoprojectors are uniformly bounded on $L^\infty$.

We identify the torus with the unit interval $[0,1)$ and let $n,\nu\in\mathbb{N}$, $1\leq \nu\leq n-1$ and $N:=n+\nu$. Then we define the (partially equally spaced) knots \[ t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{for}j=0,...,2\nu,…

In a 1973 paper Carl de Boor conjectured and 26 years later in 1999 Alexei Shadrin proved in full generality that the $L_\infty$-norm of the spline orthoprojector is bounded independently of the knot sequence for every order of the spline…

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ homotopic to the identity. A set of hypotheses is presented, called fully essential system of curves $\mathscr{C}$ and it is shown that under…

We show that $L^\infty$-norms of orthoprojectors on certain types of perturbations of spline spaces are bounded independently of the knot sequence. Explicit applications of this result are given, one of them being orthoprojectors onto…

We derive upper bounds on the difference between the orthogonal projections of a smooth function $u$ onto two finite element spaces that are nearby, in the sense that the support of every shape function belonging to one but not both of the…

We prove the analogue of the Martingale Convergence Theorem for polynomial spline sequences. Given a natural number $k $ and a sequence $(t_i)$ of knots in $[0,1]$ with multiplicity $\le k-1$, we let $P_n $ be the orthogonal projection onto…

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

The aim of this paper is to investigate the boundedness of periodic Fourier integral operators in Lebesgue spaces with variable exponent $L^{p(\cdot)}$ on the $n$-dimensional torus. We deal with operators of type $(\rho, \delta)$ which…

The main result of this paper refers to the boundedness of the orthogonal projection $P_{\alpha}:L^{2}(\mathbb{R}^{n},d\mu_{\alpha})\rightarrow \mathcal{H}_{\alpha}^{2}, n\geq2 $ associated to the harmonic Fock space…

For the L_2-orthogonal projector P onto spaces of linear splines over simplicial partitions of polyhedral domains in R^d, d>1, we show that the L_infty norm of P cannot be bounded uniformly with respect to the partition. This is in contrast…

We present an explicit formula for the orthogonal projection onto the subspace of analytic polynomials of degree at most $n$ in the local Dirichlet space $D_\mu$ , where the positive measure $\mu$ consists of a finite number of Dirac…

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\PGL_n(\R)$. After attaching to each periodic…

Let $P$ be orthogonal projection on B-splines of degree $r-1$ with equally spaced knots. Sweldens and Piessens proved that $P(x^r)-x^r$ is Bernoulli polynomial. We generalize Sweldens ans Piessens's result for box-splines. It gives the…

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in {\Bbb R}^{md}$, let $\pi^{\bf a}\colon \Sigma=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ denote…

Building on works of Berth\'e--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion.…

Fixed a continuous kernel K on the $d$-dimensional torus, we consider a generalization of the univariate $sk$-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given…