scholarly journals Functions of Boundedkthp-Variation and Continuity Modulus

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Odalis Mejía ◽  
Pilar Silvestre
Keyword(s):  
Sobolev Space ◽  
Modulus Of Smoothness ◽  
Continuity Modulus ◽  
Sharp Estimates ◽  
Class Of Functions

A scale of spaces exists connecting the class of functions of boundedkthp-variation in the sense of Riesz-Merentes with the Sobolev space of functions withp-integrablekth derivative. This scale is generated by the generalized functionals of Merentes type. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of smoothness of orderk-1/p.

On Krylov’s estimates for optional semimartingales

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
M. Abdelghani ◽  
A. Melnikov ◽  
A. Pak
Keyword(s):  
Sobolev Space ◽  
Measurable Function ◽  
Diffusion Processes ◽  
Stochastic Integrals ◽  
Controlled Diffusion ◽  
Convergence Of Solutions ◽  
Change Of Variables ◽  
General Class Of Functions ◽  
Class Of Functions ◽  
Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

scholarly journals Некоторые оценки для обобщенного преобразования Фурье, ассоциированного с оператором Чередника - Опдама

2018 ◽  
Author(s):  
H.S. Lafdal ◽  
R. Daher ◽  
El.O. Salah
Keyword(s):  
Fourier Transform ◽  
Translation Operator ◽  
Modulus Of Smoothness ◽  
Continuity Modulus ◽  
Generalized Translation Operator ◽  
Generalized Continuity ◽  
Generalized Translation ◽  
Translation Operators ◽  
Alpha Beta ◽  
Generalized Continuity Modulus

In the classical theory of approximation of functions on $\mathbb{R}^+$, the modulus of smoothness are basically built by means of the translation operators $f \to f(x+y)$. As the notion of translation operators was extended to various contexts (see [2] and [3]), many generalized modulus of smoothness have been discovered. Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [4] and [5]). In [1], Abilov et al. proved two useful estimates for the Fourier transform in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus, using a translation operator. In this paper, we also discuss this subject. More specifically, we prove some estimates (similar to those proved in [1]) in certain classes of functions characterized by a generalized continuity modulus and connected with the generalized Fourier transform associated with the differential-difference operator $T^{(\alpha,\beta)}$ in $L^{2}_{\alpha,\beta}(\mathbb{R})$. For this purpose, we use a generalized translation operator.

Sharp estimates of semistable radial solutions of k-Hessian equations

2019 ◽  
Vol 150 (4) ◽  
pp. 2083-2115 ◽  
Author(s):  
Miguel Angel Navarro ◽  
Justino Sánchez
Keyword(s):  
Sobolev Space ◽  
Weighted Sobolev Space ◽  
Extremal Solution ◽  
Nonincreasing Function ◽  
Pointwise Estimates ◽  
Radial Solutions ◽  
Sharp Estimates ◽  
Dirichlet Data ◽  
Hessian Equations ◽  
Hessian Operator

AbstractWe consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

Sharp estimates of the k -modulus of smoothness of Bessel potentials

2010 ◽  
Vol 81 (3) ◽  
pp. 608-624 ◽  
Author(s):  
Amiran Gogatishvili ◽  
Júlio S. Neves ◽  
Bohumír Opic
Keyword(s):  
Modulus Of Smoothness ◽  
Sharp Estimates ◽  
Bessel Potentials

scholarly journals Dunkl analogue of Szász-mirakjan operators of blending type

2018 ◽  
Vol 16 (1) ◽  
pp. 1344-1356 ◽  
Author(s):  
Sheetal Deshwal ◽  
P.N. Agrawal ◽  
Serkan Araci
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Integral Form ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Korovkin Theorem ◽  
Dunkl Analogue ◽  
Derivatives Of ◽  
Class Of Functions

AbstractIn the present work, we construct a Dunkl generalization of the modified Szász-Mirakjan operators of integral form defined by Pǎltanea [1]. We study the approximation properties of these operators including weighted Korovkin theorem, the rate of convergence in terms of the modulus of continuity, second order modulus of continuity via Steklov-mean, the degree of approximation for Lipschitz class of functions and the weighted space. Furthermore, we obtain the rate of convergence of the considered operators with the aid of the unified Ditzian-Totik modulus of smoothness and for functions having derivatives of bounded variation.

SHARP BOUNDS OF SOME COEFFICIENT FUNCTIONALS OVER THE CLASS OF FUNCTIONS CONVEX IN THE DIRECTION OF THE IMAGINARY AXIS

2019 ◽  
Vol 100 (1) ◽  
pp. 86-96 ◽  
Author(s):  
NAK EUN CHO ◽  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO
Keyword(s):  
Analytic Functions ◽  
Imaginary Axis ◽  
Schwarz Lemma ◽  
Hankel Determinant ◽  
Sharp Estimates ◽  
The Third ◽  
Carathéodory Functions ◽  
Unit Polydisk ◽  
Class Of Functions ◽  
The Schwarz Lemma

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

scholarly journals Geometric analysis on Cantor sets and trees

2017 ◽  
Vol 2017 (725) ◽  
Author(s):  
Anders Björn ◽  
Jana Björn ◽  
James T. Gill ◽  
Nageswari Shanmugalingam
Keyword(s):  
Sobolev Space ◽  
Besov Space ◽  
Besov Spaces ◽  
Rooted Tree ◽  
Geometric Analysis ◽  
Cantor Sets ◽  
Rooted Trees ◽  
First Order ◽  
Sharp Estimates ◽  
Type Sets

AbstractUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.

CLASS OF BOOLEAN FUNCTIONS CONSTRUCTED USING SIGNIFICANT BITS OF LINEAR RECURRENCES OVER THE RING ℤ2n

2019 ◽  
Vol 6 (2) ◽  
pp. 90-94
Author(s):  
Hernandez Piloto Daniel Humberto
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Linear Recurrences ◽  
Maximum Period ◽  
Non Linear ◽  
The Given ◽  
Class Of Functions

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

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