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About 2,030 papers for “Modulus of continuity”

A Continuous Modulus of Continuity

J. A. Guthrie · American Mathematical Monthly · 1983

(1983). A Continuous Modulus of Continuity. The American Mathematical Monthly: Vol. 90, No. 2, pp. 126-127.

Cited by 6

Modulus of continuity of operator functions

yulia Farforovskaya, Lioudmila Nikolskaïa · HAL (Le Centre pour la Communication Scientifique Directe) · 2008

A, B bounded self-adjoint operators on a separable Hilbert space, f continuous function, wf modulus of continuity of f.

Cited by 1

On L² modulus of continuity of Brownian . . .

Aurélien Deya, David Nualart, Samy Tindel · 2013

This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on 3 closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L² modulus of continuity for a 1-dimensional Brownian motion. (c) Extension of the second result to a 2-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.

Cited by 0

Modulus of continuity of the Dirichlet solutions

Hiroaki Aikawa · Bulletin of the London Mathematical Society · 2010

Let D be a bounded domain in ℝn with n ⩾ 2. For a function f on ∂D we denote by ℋD f the Dirichlet solution, for the Laplacian, of f over D. It is classical that if D is regular, then ℋD maps the family of continuous boundary functions to the family of harmonic functions in D continuous up to the boundary ∂D. We show that improved continuity of a boundary function f ensures improved continuity of ℋD f in the context of general modulus of continuity.

Cited by 0Open access

On modulus of continuity of a convolution

Susan Elizabeth Royer Burns · OakTrust (Texas A&M University Libraries) · 1969

Due to the character of the original source materials and the nature of batch digitization, quality control issues may be present in this document. Please report any quality issues you encounter to digital@library.tamu.edu, referencing the URI of the item.

Cited by 0

Luzin's condition (N) and modulus of continuity

Pekka Koskela, Jan Malý, Thomas Zürcher · Advances in Calculus of Variations · 2014

Abstract In this paper, we establish Luzin's condition (N) for mappings in certain Sobolev–Orlicz spaces with certain moduli of continuity. Further, given a mapping in these Sobolev–Orlicz spaces, we give bounds on the size of the exceptional set where Luzin's condition (N) may fail. If a mapping violates Luzin's condition (N), we show that there is a Cantor set of measure zero that is mapped to a set of positive measure.

Cited by 0Open access

Functions with a concave modulus of continuity

Helen E. White · Proceedings of the American Mathematical Society · 1974

In [<bold>1</bold>], C. Goffman proved that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a modulus of continuity, then the set of all functions <italic>f</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-parenthesis StartSet x colon f left-parenthesis x right-parenthesis equals g left-parenthesis x right-parenthesis EndSet right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>:</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m(\{ x:f(x) = g(x)\} ) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<italic>m</italic> denotes Lebesgue measure) for all <italic>g</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the set of all functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a modulus of continuity, is residual in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the present article, we prove that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a concave modulus of continuity and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than upper K greater-than 24 Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>K</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>24</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">0 &gt; K &gt; {24^{ - 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the set of all functions <italic>f</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-parenthesis StartSet x colon f left-parenthesis x right-parenthesis equals g left-parenthesis x right-parenthesis EndSet right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>:</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m(\{ x:f(x) = g(x)\} ) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <italic>g</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper K sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(K\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is residual in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using this result, we show that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than alpha greater-than 1"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0 &gt; \alpha &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then there are functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C[0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which satisfy a Hölder condition of exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α<!-- α --></mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-parenthesis StartSet x colon f left-parenthesis x right-parenthesis equals g left-parenthesis x right-parenthesis EndSet right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>:</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> <m

Cited by 0Open access

Modulus of continuity for a martingale sequence

Azat Miftakhov · arXiv (Cornell University) · 2020

Given a martingale sequence of random fields that satisfies a natural assumption of boundedness, it is shown that the pointwise limit of this sequence can be modified in such a way that a certain class of moduli of continuity is preserved. That is, if every element of the sequence admits a given modulus of continuity, one can construct a modification of the limiting random field so that this new field also admits the same modulus of continuity. Additionally, it is shown that requiring further smoothness and a stronger notion of boundedness for the original sequence guarantees further smoothness of the limiting field and a stronger mode of convergence to this limit. Moreover, the modulus of continuity is also preserved for the derivatives.

Cited by 0Open access

The boundary modulus of continuity of harmonic functions

Elgin Johnston · Pacific Journal of Mathematics · 1980

Let G be a bounded domain in the complex plane and let u(z) be continuous on G. In this paper we study the boundary modules of continuity, (), of u on dG and the modulus of continuity, (), of u on G. We investigate the extent to which the inequality "() <(<5)" holds when u is harmonic on G and show that the precise formulation of such inequalities depends on the smoothness of dG.

Cited by 30Open access

On the modulus of continuity for starlike mappings

Dieter Gaier, Reiner hnau, Jan Krzy et al. · UMCS Library (Maria Curie-Skłodowska University) · 2002

For a conformal mapping of the unit disk onto a starlike domain with boundary in a given annulus we derive an estimate for the modulus of continuity of the boundary correspondence function.The result is in some sense asymptotically sharp.

Cited by 0Open access