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About 10,000 papers for “Bounded function”

Continuous function on closed bounded sets

T. W. Körner, Terence Tao · 2022

Cited by 0

Bounded Sequence-to-Function Hausdorff Transformations

Constantine Georgakis · Proceedings of the American Mathematical Society · 1988

Let \[ \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0\] be the sequence-to-function Hausdorff transformation generated by the completely monotone function $g$ or, what is equivalent, the Laplace transform of a finite positive measure $\sigma$ on $[0,\infty )$. It is shown that for $1 \leq p \leq \infty$, $T$ is a bounded transformation of ${l^p}$ with weight $\Gamma \left ( {n + s + 1} \right ) / n!$ into ${L^p}[0,\infty )$ with weight ${y^s},s > - 1$, whose norm $\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )$ if and only if $C\left ( {p,s} \right ) < \infty$, and that for $1 < p < \infty ,{\left \| {Ta} \right \|_{p,s}} < C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}$ unless ${a_n}$ is a null sequence. Furthermore, if $1 < p < r < \infty , \;0 < \lambda < 1$ and $\sigma$ is absolutely continuous with derivatives $\psi$ such that the function ${\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )$ belongs to ${L^{1 / \lambda }}[0,\infty )$, then the transformation $\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )$ is bounded from ${l^p}$ to ${L^r}[0,\infty )$ and has norm $\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}$. The transformation $T$ includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on ${l^p}$ which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.

Cited by 1Open access

Bounded sequence-to-function Hausdorff transformations

Constantine Georgakis · Proceedings of the American Mathematical Society · 1988

Let <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T a right-parenthesis left-parenthesis y right-parenthesis equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts left-parenthesis negative y right-parenthesis Superscript n Baseline StartFraction g Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis y right-parenthesis Over n factorial EndFraction a Subscript n comma Baseline y greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo movablelimits="false">∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:munderover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:mfrac> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> </mml:mrow> </mml:msub> </mml:mrow> <mml:mspace width="1em" /> <mml:mi>y</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> be the sequence-to-function Hausdorff transformation generated by the completely monotone function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or, what is equivalent, the Laplace transform of a finite positive measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq p \leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a bounded transformation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{l^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis n plus s plus 1 right-parenthesis slash n factorial"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \left ( {n + s + 1} \right ) / n!</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript s Baseline comma s greater-than negative 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{y^s},s > - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose norm <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T double-vertical-bar equals integral Subscript 0 Superscript normal infinity Baseline t Superscript minus left-parenthesis 1 plus s right-parenthesis slash p Baseline d sigma left-parenthesis t right-parenthesis equals upper C left-parenthesis p comma s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>T</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD">

Cited by 2

Bounded influence magnetotelluric response function estimation

Alan D. Chave, David J. Thomson · Geophysical Journal International · 2004

Author Posting. © Blackwell, 2004. This article is posted here by permission of Blackwell for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 157 (2004): 988–1006, doi:10.1111/j.1365-246X.2004.02203.x.

Cited by 326Open access

Uniformly antisymmetric function with bounded range

Krzysztof Ciesielski, Saharon Shelah · arXiv (Cornell University) · 1998

The goal of this note is to construct a uniformly antisymmetric function f:R-> R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson. (See also list of problems in Thomson and Problem 2(b) from Ciesielski's survey.) A problem of existence of uniformly antisymmetric function f:R-> R with finite range remains open.

Cited by 0Open access

Continuous function on closed bounded sets

T. W. Körner · 1988

Cited by 1

The bounded complexity function versus the unbounded complexity function

S. J. Qian, Wenxiang Sun, Cheng Zhang · Journal of Mathematical Analysis and Applications · 2011

Cited by 0

Partition Function Estimation under Bounded f-Divergence

Adam Block, Abhishek Shetty · ArXiv.org · 2026

We study the statistical complexity of estimating partition functions given sample access to a proposal distribution and an unnormalized density ratio for a target distribution. While partition function estimation is a classical problem, existing guarantees typically rely on structural assumptions about the domain or model geometry. We instead provide a general, information-theoretic characterization that depends only on the relationship between the proposal and target distributions. Our analysis introduces the integrated coverage profile, a functional that quantifies how much target mass lies in regions where the density ratio is large. We show that integrated coverage tightly characterizes the sample complexity of multiplicative partition function estimation and provide matching lower bounds. We further express these bounds in terms of $f$-divergences, yielding sharp phase transitions depending on the growth rate of f and recovering classical results as a special case while extending to heavy-tailed regimes. Matching lower bounds establish tightness in all regimes. As applications, we derive improved finite-sample guarantees for importance sampling and self-normalized importance sampling, and we show a strict separation between the complexity of approximate sampling and counting under the same divergence constraints. Our results unify and generalize prior analyses of importance sampling, rejection sampling, and heavy-tailed mean estimation, providing a minimal-assumption theory of partition function estimation. Along the way we introduce new technical tools including new connections between coverage and $f$-divergences as well as a generalization of the classical Paley-Zygmund inequality.

Cited by 0Open access

Bounded Blending for Function-Based Shape Modeling

Galina Pasko, Alexander Pasko, Tosiyasu L. Kunii · IEEE Computer Graphics and Applications · 2005

New analytical formulations of bounded blending operations can enhance function-based constructive shape modeling. In this article, we introduce bounded-blending operations that we define using R-functions and displacement functions with the localized area of influence. We define the shape and location of the blend by control points on the surfaces of two solids or by an additional bounding solid. We can apply our proposed blending using a bounding solid to a single selected edge or vertex. We also introduce new multiple blends and a partial edge blend. Our description supports set-theoretic operations on blends and blends on blends - that is, recursive blends. In this sense, our proposed operations could replace pure set-theoretic operations in the construction of a solid without rebuilding the entire construction tree data structure. Our proposed blending method can have application in interactive design.

Cited by 53

Function-Correcting Codes for Locally Bounded Functions

Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti et al. · 2025

In this paper, we introduce a class of functions that assume only a limited number λ of values within a given Hamming ρ-ball and call them locally (ρ,λ)-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when λ = 4. We also demonstrate that any function can be represented as a locally (ρ,λ)-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.

Cited by 1

Bounded point derivations on certain function spaces

James E. Brennan · St Petersburg Mathematical Journal · 2020

Let $X$ be a compact nowhere dense subset of the complex plane $\mathbb {C}$, and let $dA$ denote two-dimensional Lebesgue (or area) measure in $\mathbb {C}$. Denote by $\mathcal {R}(X)$ the set of all rational functions having no poles on $X$, and by $R^p(X)$ the closure of $\mathcal {R}(X)$ in $L^p(X,dA)$ whenever $1\leq p<\infty$. The purpose of this paper is to study the relationship between bounded derivations on $R^p(X)$ and the existence of approximate derivatives provided $2<p<\infty$, and to draw attention to an anomaly that occurs when $p=2$.

Cited by 0

An Entire Function Bounded in Every Direction

David Newman · American Mathematical Monthly · 1976

"An Entire Function Bounded in Every Direction." The American Mathematical Monthly, 83(3), pp. 192–193

Cited by 12

The Bohr Spectrum of a Bounded Function

H. G. Eggleston · Proceedings of the American Mathematical Society · 1958

Cited by 1

Bounded Point Derivations on Certain Function Algebras

James E. Brennan · Operator theory · 2018

Cited by 1

Partition Function Estimation under Bounded f-Divergence

Adam Block, Abhishek Shetty · arXiv (Cornell University) · 2026

Cited by 0Open access

Multiplicity function for functions of bounded variation

Kelleher, Brother Roch · OpenBU (Boston University) · 2019

Cited by 0

Function-Correcting Codes for Locally Bounded Functions

Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti et al. · ArXiv.org · 2025

In this paper, we introduce a class of functions that assume only a limited number $λ$ of values within a given Hamming $ρ$-ball and call them locally $(ρ, λ)$-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when $λ= 4$. We also demonstrate that any function can be represented as a locally $(ρ, λ)$-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.

Cited by 0Open access

On matrix rank function over bounded arithmetics

Eitetsu Ken, Satoru Kuroda · arXiv (Cornell University) · 2023

In [Mulmuley, 1987], Mulmuley gave an algorithm reducing the computation of the matrix rank function to that of determinants, of which the proof for the verification is elementary. In this article, we formalize this argument in the bounded arithmetic $LAP$; that is, we show that \[\det(AB)=\det(A)\det(B)\] for matrices $A,B$ with $mathbb{F}(X)$-coefficients implies \[rank(M)=dim(im M),\] where $\mathbb{F}$ is the universe of the field-sort of the theory, $M$ is a matrix with $\mathbb{F}$-coefficients, and $rank(M)$ is the rank function computed by Mulmuley's algorithm. Furthermore, interpreting $LAP$ by $VNC^{2}$ with $\mathbb{F}=\mathbb{Q}$ and using the result of [Tzameret \& Cook, 2021], we see that $VNC^{2}$ can formalize $rank(M)$ and prove $rank(M)=dim(im M)$. Lastly, we give several examples of combinatorial statements provable in $VNC^{2}$, using the formalized linear algebra.

Cited by 0Open access

The Bohr spectrum of a bounded function

H. G. Eggleston · Proceedings of the American Mathematical Society · 1958

In this note it is shown that this conjecture is correct. The exceptional set may however be non-enumerable. The result is a consequence of a connection between the set of values t for which (1) is false and that for which the nonintegral parts of the sequence { nkt are not equidistributed, where nk is an increasing sequence of integers satisfying nk+1 nk < C for some constant C. DEFINITIONS AND NOTATION. The set of values of t for which (1) is false will be denoted by T(4) and any set which is a T(4) set for a bounded and uniformly continuous function O(x) will be said to be of type I. The set of values of t for which the nonintegral parts of the sequence fnkt } are not equidistributed will be denoted by T { nkIk}. If the sequence { nk } is a strictly increasing sequence of positive integers for which nk+1 nk < C where C is a constant, then the set T { nk } is said to be of type II. r will be used to denote an outer measure function in the sense of Caratheodory. If I is a real number, [1] denotes the largest integer not greater than 1. The connection between equidistribution and the property expressed in (1) is given by the following theorem

Cited by 4Open access

On separation by function of bounded variation

Andrzej Olbryś · Periodica Mathematica Hungarica · 2023

Abstract In the present paper we are concerned with the problem of separation of two given functions by an increasing (decreasing) function or by a function of bounded variation. In the context of separation by a function of bounded variation we give two approaches. The first one concerns the separation with respect to the classical order in the set of real numbers, and this allow us to the new concept of variation involving two functions. This approach in fact leads us to the unusual phenomenon, namely the bounded joint variation of two functions forces these functions to be essentially the same. The second approach concerns the separation with respect to the partial order generated by the Lorentz cone.

Cited by 0Open access