Recent zbMATH articles in MSC 41A15 https://www.zbmath.org/atom/cc/41A15 2022-05-16T20:40:13.078697Z Werkzeug Extremal problems for non-periodic splines on real domain and their derivatives https://www.zbmath.org/1483.41002 2022-05-16T20:40:13.078697Z "Danchenko, K. A." https://www.zbmath.org/authors/?q=ai:danchenko.k-a "Kofanov, V. A." https://www.zbmath.org/authors/?q=ai:kofanov.v-a Summary: We consider the Bojanov-Naidenov problem over the set $$\sigma_{h,r}$$ of all non-periodic splines $$s$$ of order $$r$$ and minimal defect with knots at the points $$kh, k \in \mathbb{Z}$$. More exactly, for given $$n, r \in \mathbb{N}$$; $$p, A > 0$$ and any fixed interval $$[a, b] \subset \mathbb{R}$$ we solve the following extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup,\quad q \geqslant p,\tag{1}$ over the classes $\sigma_{h,r}^p(A) := \left\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \quad \delta \in (0, h], \quad \tau \in \mathbb{R} \right\},$ where $\| x \|_{p, \delta} := \sup \left\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, \quad 0 < b - a \leqslant \delta \right\},$ and $$\varphi_{\lambda, r}$$ is a $$(2\pi / \lambda)$$-periodic spline of Euler of order $$r$$. In particularly, for $$k = 1, \dots, r - 1$$ we solve the extremal problem $\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup,\quad q \geqslant 1, \tag{2}$ over the classes $$\sigma_{h,r}^p (A)$$. Note that the problems (1) and (2) were solved earlier on the classes $\sigma_{h,r}(A, p) := \left\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \quad \tau \in \mathbb{R} \right\},$ where $L(x)_p := \sup \left\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \right\}.$ We prove that the classes $$\sigma_{h,r}^p (A)$$ are wider than the classes $$\sigma_{h,r}(A,p)$$. Similarly, we solve the analog of Erdös problem about the characterization of the spline $$s \in \sigma_{h,r}^p(A)$$ that has maximal arc length over a fixed interval $$[a, b] \subset \mathbb{R}$$. The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines https://www.zbmath.org/1483.41003 2022-05-16T20:40:13.078697Z "Asadova, E. V." https://www.zbmath.org/authors/?q=ai:asadova.e-v "Kofanov, V. A." https://www.zbmath.org/authors/?q=ai:kofanov.v-a Summary: For given $$n, r \in \mathbb{N}$$; $$p, A > 0$$ and any fixed interval $$[a,b] \subset \mathbb{R}$$ we solve the extremal problem $$\int\limits_a^b |x(t)|^q dt \rightarrow \sup$$, $$q \geqslant p$$, over sets of trigonometric polynomials $$T$$ of order $$\leqslant n$$ and $$2\pi$$-periodic splines $$s$$ of order $$r$$ and minimal defect with knots at the points $$k\pi / n$$, $$k \in \mathbb{Z}$$, such that $$\| T \|_{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$$, $$\delta \in (0, \pi / n]$$, where $$\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$$ and $$\varphi_{n, r}$$ is the $$(2\pi / n)$$-periodic spline of Euler of order $$r$$. In particular, we solve the same problem for the intermediate derivatives $$x^{(k)}$$, $$k = 1, \dots, r-1$$, with $$q \geqslant 1$$. Signal analysis using Born-Jordan-type distributions https://www.zbmath.org/1483.42004 2022-05-16T20:40:13.078697Z "Cordero, Elena" https://www.zbmath.org/authors/?q=ai:cordero.elena "de Gosson, Maurice" https://www.zbmath.org/authors/?q=ai:de-gosson.maurice-a "Dörfler, Monika" https://www.zbmath.org/authors/?q=ai:dorfler.monika "Nicola, Fabio" https://www.zbmath.org/authors/?q=ai:nicola.fabio The Chapter contains results concerning recent advances in signal theory using time-frequency distributions. The authors demonstrate that some new members of the Cohen class that generalize the Wigner distribution are useful in damping artefacts in certain signals. Main properties as well as drawbacks are presented. Last but not least, several open problems are also discussed. For the entire collection see [Zbl 1470.42002]. Reviewer: Liviu Goraş (Iaşi) Variational time discretization of Riemannian splines https://www.zbmath.org/1483.49017 2022-05-16T20:40:13.078697Z "Heeren, Behrend" https://www.zbmath.org/authors/?q=ai:heeren.behrend "Rumpf, Martin" https://www.zbmath.org/authors/?q=ai:rumpf.martin "Wirth, Benedikt" https://www.zbmath.org/authors/?q=ai:wirth.benedikt Summary: We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy -- a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity -- under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the $$\Gamma$$-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing and on the infinite-dimensional shape manifold of viscous rods. Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis https://www.zbmath.org/1483.65026 2022-05-16T20:40:13.078697Z "Sande, Espen" https://www.zbmath.org/authors/?q=ai:sande.espen "Manni, Carla" https://www.zbmath.org/authors/?q=ai:manni.carla "Speleers, Hendrik" https://www.zbmath.org/authors/?q=ai:speleers.hendrik Isogeometric analysis [\textit{T. J. R. Hughes} et al., Comput. Methods Appl. Mech. Eng. 194, No. 39--41, 4135--4195 (2005; Zbl 1151.74419)] is a generalization of the classical finite element analysis, and it advocates for the use of splines as finite elements. This has led to a renewed interest in using splines for approximation, and numerical evidence has shown the benefits of their smoothness -- smooth spline spaces demonstrate better approximation behaviour per degree of freedom than less smooth spline spaces. The authors of the present manuscript provide a priori error estimates that, in particular, help to describe the afore-mentioned numerical evidence. The error estimates are for $$L^2$$ and Ritz projections onto spline spaces of arbitrary smoothness defined on non-uniform meshes, and they feature constants that depend explicitly on the spline space parameters -- mesh size $$h$$, polynomial degree $$p$$ and smoothness $$k$$. The key to their results is the description of the considered Sobolev spaces and spline spaces in terms of integral operators what allows them to derive explicit constants in spline approximation from those in polynomial approximation. The error estimates are provided for univariate spline spaces, reduced univariate spline spaces (i.e., spline spaces with zero end-point even or odd derivatives) and multivariate tensor-product spline spaces. The multivariate results cover approximation on the spline parametric domain as well as mapped single-patch and $$C^0$$ multi-patch geometries; the role of the geometric map is explicitized in the constants. Finally, the error estimates improve upon the ones presented earlier in [\textit{L. Beirão da Veiga} et al., Numer. Math. 118, No. 2, 271--305 (2011; Zbl 1222.41010); \textit{S. Takacs} and \textit{T. Takacs}, Math. Models Methods Appl. Sci. 26, No. 7, 1411--1445 (2016; Zbl 1339.41012)] and [\textit{E. Sande} et al., Math. Models Methods Appl. Sci. 29, No. 6, 1175--1205 (2019; Zbl 1428.41010)]. Reviewer: Deepesh Toshniwal (Delft)