scholarly journals Linear Interpolation on a Euclidean Ball in Rⁿ

2019 ◽  
Vol 26 (2) ◽  
pp. 279-296
Author(s):  
Mikhail V. Nevskii ◽  
Alexey Yu. Ukhalov
Keyword(s):  
Linear Interpolation ◽  
Continuous Functions ◽  
Linear Functions ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Integer Number ◽  
Space Of Continuous Functions ◽  
Lagrange Polynomials ◽  
Computational Data ◽  
Theoretical Results

For \(x^{(0)}\in{\mathbb R}^n, R>0\), by \(B=B(x^{(0)};R)\) we denote a Euclidean ball in \({\mathbb R}^n\) given by~the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}\). Put \(B_n:=B(0,1)\). We mean by \(C(B)\) the space of~continuous functions \(f:B\to{\mathbb R}\) with the norm \(\|f\|_{C(B)}:=\max_{x\in B}|f(x)|\) and by \(\Pi_1\left({\mathbb R}^n\right)\) the set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions on \({\mathbb R}^n\). Let \(x^{(1)}, \ldots, x^{(n+1)}\) be the~vertices of \(n\)-dimensional nondegenerate simplex \(S\subset B\). The interpolation projector \(P:C(B)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).\) Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) making use of \(x^{(0)}\), \(R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \(B_n\). In this situation, we prove that \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|\) \((0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \(\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}\). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

scholarly journals On Some Problems for a Simplex and a Ball in Rn

2018 ◽  
Vol 25 (6) ◽  
pp. 680-691
Author(s):  
Mikhail V. Nevskii
Keyword(s):  
Convex Body ◽  
Geometric Interpretation ◽  
Unit Cube ◽  
Linear Functions ◽  
Barycentric Coordinates ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Lagrange Polynomials ◽  
Dimensional Unit ◽  
Unit Euclidean Ball

Let \(C\) be a convex body and let \(S\) be a nondegenerate simplex in \({\mathbb R}^n\). Denote by \(\tau S\) the image of \(S\) under homothety with a center of homothety in the center of gravity of \(S\) and the ratio \(\tau\). We mean by \(\xi(C;S)\) the minimal \(\tau>0\) such that \(C\) is a subset of the simplex \(\tau S\). Define \(\alpha(C;S)\) as the minimal \(\tau>0\) such that \(C\) is contained in a translate of \(\tau S\). Earlier the author has proved the equalities \(\xi(C;S)=(n+1)\max\limits_{1\leq j\leq n+1}\max\limits_{x\in C}(-\lambda_j(x))+1\)  (if \(C\not\subset S\)), \(\alpha(C;S)=\sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-\lambda_j(x))+1.\)Here \(\lambda_j\) are the linear functions that are called the basic Lagrange polynomials corresponding to \(S\). The numbers \(\lambda_j(x),\ldots, \lambda_{n+1}(x)\) are the barycentric coordinates of a point \(x\in{\mathbb R}^n\). In his previous papers, the author investigated these formulae in the case when \(C\) is the \(n\)-dimensional unit cube \(Q_n=[0,1]^n\). The present paper is related to the case when \(C\) coincides with the unit Euclidean ball \(B_n=\{x: \|x\|\leq 1\},\) where \(\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.\) We establish various relations for \(\xi(B_n;S)\) and \(\alpha(B_n;S)\), as well as we give their geometric interpretation. For example, if \(\lambda_j(x)=l_{1j}x_1+\ldots+l_{nj}x_n+l_{n+1,j},\) then \(\alpha(B_n;S)=\sum\limits_{j=1}^{n+1}\left(\sum\limits_{i=1}^n l_{ij}^2\right)^{1/2}\). The minimal possible value of each characteristics \(\xi(B_n;S)\) and \(\alpha(B_n;S)\) for \(S\subset B_n\) is equal to \(n\). This value corresponds to a regular simplex inscribed into \(B_n\). Also we compare our results with those obtained in the case \(C=Q_n\).

scholarly journals On Properties of a Regular Simplex Inscribed into a Ball

2021 ◽  
Vol 28 (2) ◽  
pp. 186-197
Author(s):  
Mikhail Viktorovich Nevskii
Keyword(s):  
Uniform Norm ◽  
Linear Functions ◽  
Regular Simplex ◽  
Euclidean Ball ◽  
Minimal Norm ◽  
Lagrange Polynomials

Let  $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector  $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$,  $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$.  We prove that this conjecture holds true at least for $n=1,2,3,4$. 

scholarly journals On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

2018 ◽  
Vol 25 (3) ◽  
pp. 291-311
Author(s):  
Mikhail V. Nevskii ◽  
Alexey Yu. Ukhalov
Keyword(s):  
Hadamard Matrix ◽  
Continuous Functions ◽  
Unit Cube ◽  
Linear Functions ◽  
Optimal Interpolation ◽  
Maximum Volume ◽  
Space Of Continuous Functions ◽  
Lagrange Polynomials ◽  
Geometric Character ◽  
Dimensional Cube

Let \(n\in{\mathbb N}\), and let \(Q_n\) be the unit cube \([0,1]^n\). By \(C(Q_n)\) we denote the space of continuous functions \(f:Q_n\to{\mathbb R}\) with the norm \(\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,\) by \(\Pi_1\left({\mathbb R}^n\right)\) --- the set of polynomials of \(n\) variables of degree \(\leq 1\) (or linear functions). Let \(x^{(j)},\) \(1\leq j\leq n+1,\) be the vertices of \(n\)-dimnsional nondegenerate simplex \(S\subset Q_n\). An interpolation projector \(P:C(Q_n)\to \Pi_1({\mathbb R}^n)\) corresponding to the simplex \(S\) is defined by equalities \(Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).\) The norm of \(P\) as an operator from \(C(Q_n)\) to \(C(Q_n)\) may be calculated by the formula \(\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.\) Here \(\lambda_j\) are the basic Lagrange polynomials with respect to \(S,\) \(ver(Q_n)\) is the set of vertices of \(Q_n\). Let us denote by \(\theta_n\) the minimal possible value of \(\|P\|.\) Earlier, the first author proved various relations and estimates for values \(\|P\|\) and \(\theta_n\), in particular, having geometric character. The equivalence \(\theta_n\asymp \sqrt{n}\) takes place. For example, the appropriate, according to dimension \(n\), inequalities may be written in the form \linebreak \(\frac{1}{4}\sqrt{n}\) \(<\theta_n\) \(<3\sqrt{n}.\) If the nodes of the projector \(P^*\) coincide with vertices of an arbitrary simplex with maximum possible volume, we have \(\|P^*\|\asymp\theta_n.\)When an Hadamard matrix of order \(n+1\) exists, holds \(\theta_n\leq\sqrt{n+1}.\) In the paper, we give more precise upper bounds of numbers \(\theta_n\) for \(21\leq n \leq 26\). These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements \(\pm 1.\) Also, we systematize and comment the best nowaday upper and low estimates of numbers \(\theta_n\) for a concrete \(n.\)

CONTINUOUS PIECEWISE LINEAR FUNCTIONS

2005 ◽  
Vol 10 (1) ◽  
pp. 77-99 ◽  
Author(s):  
CHARALAMBOS D. ALIPRANTIS ◽  
DAVID HARRIS ◽  
RABEE TOURKY
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Linear Functions ◽  
Dimensional Euclidean Space ◽  
Piecewise Linear Functions ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Mathematical Background ◽  
Linear Regressions ◽  
Special Case

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

scholarly journals Geometric Estimates in Interpolation on an n-Dimensional Ball

2019 ◽  
Vol 26 (3) ◽  
pp. 441-449
Author(s):  
Mikhail V. Nevskii
Keyword(s):  
Legendre Polynomial ◽  
Polynomial Interpolation ◽  
Linear Interpolation ◽  
Continuous Functions ◽  
Unit Cube ◽  
Linear Functions ◽  
Dimensional Unit ◽  
Euclidean Unit Ball ◽  
Dimensional Ball ◽  
Geometric Estimates

Suppose \(n\in {\mathbb N}\). Let \(B_n\) be a Euclidean unit ball in \({\mathbb R}^n\) given by the inequality \(\|x\|\leq 1\), \(\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}\). By \(C(B_n)\) we mean a set of continuous functions \(f:B_n\to{\mathbb R}\) with the norm \(\|f\|_{C(B_n)}:=\max\limits_{x\in B_n}|f(x)|\). The symbol \(\Pi_1\left({\mathbb R}^n\right)\) denotes a set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions upon \({\mathbb R}^n\). Assume that \(x^{(1)}, \ldots, x^{(n+1)}\) are vertices of an \(n\)-dimensional nondegenerate simplex \(S\subset B_n\). The interpolation projector \(P:C(B_n)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).\) Denote by \(\|P\|_{B_n}\) the norm of \(P\) as an operator from \(C(B_n)\) on to \(C(B_n)\). Let us define \(\theta_n(B_n)\) as the minimal value of \(\|P\|_{B_n}\) under the condition \(x^{(j)}\in B_n\). We describe the approach in which the norm of the projector can be estimated from the bottom through the volume of the simplex. Let \(\chi_n(t):=\frac{1}{2^nn!}\left[ (t^2-1)^n \right] ^{(n)}\) be the standardized Legendre polynomial of degree \(n\). We prove that \(\|P\|_{B_n}\geq\chi_n^{-1}\left(\frac{vol(B_n)}{vol(S)}\right).\) From this, we obtain the equivalence \(\theta_n(B_n)\) \(\asymp\) \(\sqrt{n}\). Also we estimate the constants from such inequalities and give the comparison with the similar relations for linear interpolation upon the \(n\)-dimensional unit cube. These results have applications in polynomial interpolation and computational geometry.

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

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