scholarly journals Approximation of functions by means of Lipschitz functions

1963 ◽  
Vol 3 (2) ◽  
pp. 134-150 ◽  
Author(s):  
J. H. Michael
Keyword(s):  
Lipschitz Function ◽  
Unit Cube ◽  
Lipschitz Functions ◽  
Approximation Of Functions ◽  
Image Position ◽  
Elementary Area

Let Q denote the closed unit cube in Rn. The elementary area A(f) of a Lipschitz function f on Q is given by the formula.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (03) ◽  
pp. 640-654
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres K n of some volume V(K n ) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form L n (s) molecules with exactly s atoms K n. The random variable L n(s) has a limit distribution if V(K n ) tends to zero but nV(Kn ) tends to infinity at a certain rate: it is shown that for L n(s) is asymptotically Poisson.

scholarly journals Error bounds in the approximation of functions

1972 ◽  
Vol 6 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Badri N. Sahney ◽  
V. Venu Gopal Rao
Keyword(s):  
Trigonometric Polynomial ◽  
Error Bounds ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Image Position

Let f(x) ε Lipα, 0 < α < 1, in the range (-π, π), and periodic with period 2π, outside this range. Also let.We define the norm asand let the degree of approximation be given bywhere Tn (x) is some n–th trigonometric polynomial.

The asymptotic distribution of random molecules

1980 ◽  
Vol 12 (3) ◽  
pp. 640-654 ◽  
Author(s):  
Wulf Rehder
Keyword(s):  
Asymptotic Distribution ◽  
Limit Distribution ◽  
Random Variable ◽  
Unit Cube ◽  
Image Position ◽  
Solid Spheres

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for Ln(s) is asymptotically Poisson.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

2020 ◽  
Vol 102 (2) ◽  
pp. 186-195
Author(s):  
WEILIANG WANG ◽  
LU LI
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Continuous Functions ◽  
Lipschitz Functions ◽  
Image Position

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set $$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$ where $\unicode[STIX]{x1D70F}_{1}$, $\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with $\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all $x,y\in [0,1]$.

Approximation of Functions by a Bernstein-Type Operator

1972 ◽  
Vol 15 (4) ◽  
pp. 551-557 ◽  
Author(s):  
S. P. Pethe ◽  
G. C. Jain
Keyword(s):  
Binomial Distribution ◽  
Type Operator ◽  
Bernstein Operator ◽  
Bernstein Type ◽  
Approximation Of Functions ◽  
Image Position

Various generalizations of the Bernstein operator, defined on C[0, 1] by the relation1.1wherehave been given. Note that bnk(x) is the well-known binomial distribution.

scholarly journals Lipschitz functions with unexpectedly large sets of nondifferentiability points

2005 ◽  
Vol 2005 (4) ◽  
pp. 361-373 ◽  
Author(s):  
Marianna Csörnyei ◽  
David Preiss ◽  
Jaroslav Tišer
Keyword(s):  
Lipschitz Function ◽  
Measure Zero ◽  
Common Point ◽  
Lipschitz Functions ◽  
Large Sets ◽  
Set Of Points ◽  
Dense Set

It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.

scholarly journals An Integralgeometric Approach to Dorronsoro Estimates

2019 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Function ◽  
Parabolic Problem ◽  
Elementary Proof ◽  
Lipschitz Functions ◽  
Geometric Proof ◽  
Proof Technique ◽  
Almost Everywhere ◽  
Small Scales ◽  
Affine Functions ◽  
Similar Proof

Abstract A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro’s theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher’s theorem for parabolic Lipschitz functions.

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 257-263 ◽  
Author(s):  
GERALD BEER ◽  
M. I. GARRIDO
Keyword(s):  
Metric Space ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
The Family ◽  
Locally Lipschitz

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

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