Operator Theory and Harmonic Analysis
OTHA 2020, Part I – New General Trends and Advances of the Theory
- 2021
- Book
- Editors
- Alexey N. Karapetyants
- Vladislav V. Kravchenko
- Elijah Liflyand
- Helmuth R. Malonek
- Book Series
- Springer Proceedings in Mathematics & Statistics
- Publisher
- Springer International Publishing
About this book
This is the first in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University in Rostov-on-Don, Russia. This volume is focused on general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multiparameter objects required when considering operators and objects with variable parameters.
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Table of Contents
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Frontmatter
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Weighted Hadamard–Bergman Convolution Operators
Smbat A. Aghekyan, Alexey N. KarapetyantsThe chapter delves into the extension of Hadamard–Bergman convolution operators to include weight parameters, which significantly impacts the formulas and conclusions. It introduces the weighted operators of fractional integration and differentiation as important examples and explores their mapping properties in weighted Lebesgue and Hölder spaces. The text is comprehensive, providing detailed proofs and highlighting the naturalness of considering weighted convolutions in this context. Additionally, it discusses potential future research directions related to Bergman type operators and generalized holomorphic functions.AI Generated
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AbstractFollowing the ideas of the recent paper by Karapetiants and Samko (Hadamard–Bergman convolution operators. Complex analysis operator theory) we extend the introduced in the mentioned paper notion of Hadamard–Bergman convolution operators to a weighted settings. We treat operators of fractional integration and differentiation as important examples of operators in the above mentioned class, and study mapping properties of certain generalized potentials in generalized Hölder spaces. -
On Initial Extensions of Mappings
A. B. Antonevich, C. DolicaninThe chapter explores the construction of initial extensions of mappings in topological spaces, focusing on different types of extensions such as prolongations and extensions. It introduces the concept of a mapping's prolongation and continuation, highlighting specific examples like analytic continuation, differentiation in distribution theory, and the multiplication of distributions. The text also discusses the closure of non-closable mappings and the joint closure of a family of mappings, providing a detailed description of the construction of these spaces using sequences and equivalence relations. The chapter concludes by introducing the Gelfand extension of a family of mappings, showcasing its application in Banach algebras and the space of maximal ideals. Throughout, the text emphasizes the importance of these constructions in various mathematical settings and their relevance to broader research in functional analysis and topology.AI Generated
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AbstractWe introduce a general concept of an extension for a mapping f : X 0 → Y , where X and Y are topological spaces and X 0 ⊂ X. Three constructions of such extensions are proposed and the corresponding examples are given. In one of them, the extension coincides with the Gelfand transform. The peculiarity of these constructions is that the domain of the extended transformation does not belong to X but is a bundle over a subset of X. -
On Multidimensional Integral Operators with Homogeneous Kernels in Classes with Asymptotics
O. G. AvsyankinThe chapter delves into the study of multidimensional integral operators with homogeneous kernels of degree (-n), focusing on their invariance under rotations. It introduces a new class of functions with specific asymptotic behavior near zero and shows that this class is invariant under the considered integral operators. The results presented have potential applications in the study of asymptotics of solutions to integral equations. The chapter builds upon the foundational work of L. G. Mikhailov and N. K. Karapetiants, contributing to the active development of the theory of integral operators with homogeneous kernels.AI Generated
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AbstractWe define a special class of functions having a given asymptotic behavior in a neighborhood of zero. It is proved that this class is invariant under multidimensional integral operators with homogeneous kernels. -
A Dirichlet Problem for Non-elliptic Equations and Chebyshev Polynomials
A. H. BabayanThe chapter discusses the Dirichlet problem for higher-order non-elliptic equations within a unit disk in the complex plane. It focuses on the characteristic equation roots and their impact on the ellipticity of the equation. The use of Chebyshev polynomials is highlighted as a novel approach to solving these equations, particularly for fourth-order equations with real roots. The text explores different cases, including when roots are real and complex, and presents theorems that establish the conditions for the solvability of the Dirichlet problem. The chapter also provides explicit solutions and solvability conditions, making it a valuable resource for researchers in the field.AI Generated
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AbstractWe consider the Dirichlet problem for the linear non-elliptic fourth order partial differential equation in the unit disk. It supposed that in the equation only fourth order terms and coefficients are constant. The solvability conditions of in-homogeneous problem and the solutions of the corresponding homogeneous problem are determined in explicit form. The solutions are obtained in the form of expansions by Chebyshev polynomials. -
Martingale Hardy-Amalgam Spaces: Atomic Decompositions and Duality
Justice Sam Bansah, Benoît F. SehbaThis chapter delves into the rich theory of martingales, tracing its origins to J. L. Doob's seminal work and subsequent developments by notable mathematicians. It highlights the versatile applications of martingale theory in Fourier analysis, complex analysis, and classical Hardy spaces, including the probabilistic proof of the Riesz Theorem and the martingale techniques in harmonic analysis. The chapter focuses on atomic decompositions and duality within martingale Hardy-amalgam spaces, offering a deep understanding of these structures and their properties. By exploring these topics, the chapter provides valuable insights into the equivalences of function spaces and the dualities that simplify the study of their properties.AI Generated
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AbstractIn this paper, we introduce the notion of martingale Hardy-amalgam spaces: \( H^s_{p,q},\,\,\mathcal {Q}_{p,q}\) and \(\mathcal {P}_{p,q}\). We present two atomic decompositions for these spaces. The dual space of \(H^s_{p,q}\) for 0 < p ≤ q ≤ 1 is shown to be a Campanato-type space. -
Robust Estimation of European and Asian Options
G. I. Beliavsky, N. V. Danilova, A. D. LogunovThe chapter focuses on the robust estimation of European and Asian options in a (B, S)-market with risky and risk-free assets. It introduces the concept of the fair price range and formulates the problem as a minimax or maximin optimization. The text differentiates itself by employing dynamic programming for solving robust optimization problems, unlike previous works that rely on linear programming. The use of unsupervised learning to estimate model parameters and the application of a two-interval model are highlighted. The chapter also includes a computational experiment comparing the two-interval model with the Cox-Ross-Rubinstein (CRR) model, providing analytical formulas and recurrent equations for markov and Asian options. The final section includes conclusions and additional references, making the chapter a comprehensive resource for professionals in financial mathematics.AI Generated
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AbstractThe statistical problem of calculating upper and lower bounds for fair prices of European and Asian options within the classic Cox-Ross-Rubinstein (CRR) model with uncertain parameters is considered. The method proposed in this paper includes statistical data analysis for determining the ranges of the model parameters. Optimal portfolios are calculated simultaneously with the upper and lower bounds. -
On Reflexivity and Other Geometric Properties of Morrey Spaces
Evgenii I. BerezhnoiThe chapter delves into the essential geometric properties of Morrey spaces, a crucial topic in harmonic analysis and the study of partial differential equations. It establishes necessary and sufficient conditions for the local Morrey space to possess properties such as the absolutely continuous norm, the Fatou property, and reflexivity. These conditions are expressed in terms of ideal function spaces X and sequence spaces l. A key aspect of the chapter is the role of the dual space representation theorem in describing the criterion for reflexivity. Additionally, the chapter presents a criterion for the Fatou property of global Morrey spaces and illustrates with an example how the absolute continuity of the norm depends on other properties of X and l. This work not only extends existing results but also offers valuable insights into the classical Morrey spaces, making it a significant contribution to the field.AI Generated
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AbstractWe propose necessary and sufficient conditions for the local Morrey space to have one of the following properties: the absolutely continuous norm, the Fatou property, the reflexivity. We propose conditions for the global Morrey space \(GM_ {l, X}^\tau \) to have the Fatou property. We give an example of global Morrey spaces \( GM_{l,X}^ \tau \), which do have not an absolutely continuous norm, and spaces l and X have an absolutely continuous norm. -
Justification of the Averaging Method for a System with Multipoint Boundary Value Conditions
D. Bigirindavyi, V. B. LevenshtamThe chapter delves into the application of the Krylov-Bogolyubov averaging method to multipoint boundary value problems, a topic that has not been extensively studied. It compares the approaches used in previous works, highlighting the use of the implicit function theorem to bypass assumptions about the solvability of perturbed problems. The main results involve defining the problem on a set Π and considering an m-point boundary value problem governed by a system with high-frequency oscillations. The chapter provides a detailed proof and analysis, culminating in the definition of an operator in a Hölder space, showcasing the method's robustness and applicability to complex systems.AI Generated
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AbstractThe Krylov-Bogolyubov averaging method is justified for normal systems of ordinary differential equations with multipoint boundary value conditions. -
Spanne-Guliyev Type Characterization for Fractional Integral Operator and Its Commutators in Generalized Orlicz–Morrey Spaces on Spaces of Homogeneous Type
Fatih DeringozThis chapter focuses on the Spanne-Guliyev type characterization for fractional integral operators and their commutators in generalized Orlicz–Morrey spaces on spaces of homogeneous type. It begins by introducing the necessary definitions and conditions for Young functions and Orlicz spaces, and then explores the boundedness of fractional integral operators and commutators in these spaces. The chapter presents detailed theorems and lemmas that establish necessary and sufficient conditions for the boundedness of these operators, highlighting the role of conditions such as Δ2 and ∇2 for Young functions. Additionally, it provides examples and remarks that contextualize the findings within existing literature, making it a valuable resource for researchers in functional analysis and harmonic analysis.AI Generated
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AbstractThis paper establishes necessary and sufficient condition for the Spanne-Guliyev type boundedness of the fractional integral operator and its commutators in generalized Orlicz–Morrey spaces over spaces of homogeneous type which satisfy the Q-homogeneous (Ahlfors regular) condition. -
Spanne Type Characterization of Parabolic Fractional Maximal Function and Its Commutators in Parabolic Generalized Orlicz–Morrey Spaces
V. S. Guliyev, A. Eroglu, G. A. AbasovaThe chapter delves into the theory of boundedness of classical operators in real analysis, such as the fractional maximal operator and Riesz potential, and their applications in partial differential equations. It focuses on the role of Orlicz spaces and their generalizations, namely generalized Orlicz–Morrey spaces, in this context. The text defines the parabolic generalized Orlicz–Morrey space of the third kind and establishes necessary and sufficient conditions for the boundedness of the parabolic fractional maximal function and its commutators in these spaces. It also provides local estimates for the parabolic fractional maximal operator and commutators, highlighting the importance of these spaces in the study of partial differential equations.AI Generated
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AbstractIn this paper, we shall give necessary and sufficient conditions for the Spanne type boundedness of the parabolic fractional maximal operator and its commutators on parabolic generalized Orlicz–Morrey spaces, res-pectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators. -
Eigenvalues of Tridiagonal Hermitian Toeplitz Matrices with Perturbations in the Off-diagonal Corners
Sergei M. Grudsky, Egor A. Maximenko, Alejandro Soto-GonzálezThe chapter focuses on the eigenvalues of n x n matrices A_α,n with complex parameter α, which belong to the classes of periodic Jacobi, Hermitian Toeplitz, and perturbed tridiagonal Toeplitz matrices. For α = 0, the matrix is a well-studied tridiagonal Toeplitz matrix, and explicit eigenvalue formulas are provided. For general α, the characteristic polynomial is expressed in terms of Chebyshev polynomials, and eigenvalues are computed explicitly for |α| = 1 or α = 0. For |α| ≠ 1, the localization of eigenvalues is described, and asymptotic formulas are derived using trigonometric or hyperbolic changes of variable. Strong perturbations (|α| > 1) are shown to have distinct behaviors, with extreme eigenvalues potentially leaving the interval [0, 4]. The chapter also introduces asymptotic expansions for the extreme eigenvalues and efficient approximations for large n, highlighting the unique spectral gap behavior of these matrices.AI Generated
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AbstractIn this paper we study the asymptotic behavior of the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, …, 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. This matrix family is a particular case of periodic Jacobi matrices. If |α|≤ 1, then the eigenvalues belong to [0, 4] and are asymptotically distributed as the function \(g(x)=4\sin ^2(x/2)\) on [0, π]. The situation changes drastically when |α| > 1 and n tends to infinity. For this case, we prove that the two extreme eigenvalues (the minimal and the maximal one) lay out of [0, 4] and converge rapidly to certain limits determined by the value of α, whilst all others belong to [0, 4] and are asymptotically distributed as g. In all cases, we derive asymptotic formulas for the eigenvalues and transform the characteristic equation to a form convenient to solve by numerical methods. -
Some Properties of the Kernel of a Transmutation Operator
T. N. Harutyunyan, A. M. TonoyanThe chapter 'Some Properties of the Kernel of a Transmutation Operator' delves into the Sturm-Liouville problem, focusing on the kernel P(x, t) and its integral equation, known as the Gelfand-Levitan equation. It explores the unique properties of P(x, t) and its representation in terms of eigenfunctions, providing a thorough analysis of spectral data and its application to the inverse problem. The text offers a deep dive into the mathematical intricacies of transmutation operators, making it a valuable resource for specialists in the field.AI Generated
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AbstractIt is known, that there exist the continuous function P(x, t), which generates the transmutation operatortransforming the solution \(\dfrac {\sin \lambda x}{\lambda }\) of − y″ = λ 2 y with initial conditions y(0) = 0, y′(0) = 1 to the solution φ(x, λ) of equation − y″ + q(x)y = λ 2 y with the same initial values. We have proved that P(x, t) is the solution of integral equation (analogue of Gelfand-Levitan equation).$$\displaystyle \varphi (x,\lambda )=\dfrac {\sin \lambda x}{\lambda }+\int _{0}^{x} P(x,s) \dfrac {\sin \lambda t}{\lambda } dt, $$where$$\displaystyle P(x,t)+F(x,t)+\int _{0}^{x} P(x,s) F(s,t)ds=0, $$where \(\lambda _n^2\) and a n are the spectral data of corresponding Sturm-Liouville problem.$$\displaystyle F(x,t)=\sum _{n=0}^{\infty } \left (\dfrac {1}{a_n}\dfrac {\sin \lambda _n x \sin \lambda _n t}{\lambda _n^2}-\dfrac {2}{\pi } \sin \left (n+\dfrac {1}{2}\right )x \sin \left (n+\dfrac {1}{2}\right )t\right ), $$ -
On a Riemann Boundary Value Problem with Infinite Index in the Half-plane
Hrachik M. Hayrapetyan, Smbat A. Aghekyan, Artavazd D. OhanyanThe chapter investigates the Riemann boundary value problem with infinite index in the complex half-plane, focusing on the classes of analytic functions that satisfy specific conditions. It explores the problem in the contexts of Cα and Lp classes, offering detailed proofs and estimates that contribute to a deeper understanding of these functions. The main results include theorems that establish conditions for the convergence and solutions of the homogeneous problem R, providing valuable insights into this complex mathematical topic.AI Generated
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AbstractThe paper considers the Riemann boundary value problem in the half-plane in the space L 1(ρ). The weight function ρ(x) has infinite number of zeros on the real axis. The boundary condition is understood in the sense of L 1(ρ). A necessary and sufficient condition is obtained for the normal solvability of the considered problem. The solutions are represented in explicit form. -
Repeated Distances and Dot Products in Finite Fields
Alex Iosevich, Charles WolfThis chapter delves into the study of repeated distances and dot products in finite fields, extending the Euclidean distance problem to this context. It defines analogous quantities, such as α(E) and β(E), representing the largest subsets with no repeated distances or dot products. The paper presents theorems providing lower bounds for these quantities in general dimensions and specific cases, such as when q is prime. Notably, it offers improved bounds for dimensions two and higher, and constructs sets matching these bounds. The findings have implications for understanding the structure of finite fields and their applications in various mathematical and computational problems.AI Generated
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AbstractLet \({\Bbb F}_q^d\), d ≥ 2, where 𝔽q is the field with q elements. Letand$$\displaystyle \Delta (E)=\{||x-y||: x,y \in E\}, \ ||x||=x_1^2+\dots +x_d^2, $$$$\displaystyle \prod (E)=\{x \cdot y: x,y \in E\}, \ x \cdot y=x_1y_1+\dots +x_dy_d. $$The purpose of this paper is to find the largest possible subset E′ of E such that all the distances determined by E′ are distinct, and also find a subset E″ of E such that all the dot products determined by E″ are distinct. We provide some number theoretic examples that indicate the degree of sharpness of our results. A general mechanism is outlined that should allow one to study these problems in much greater generality. -
Banach Spaces of Functions Delta-Subharmonic in the Unit Disc
Armen Jerbashian, Jesus PejendinoThe chapter delves into the intricate study of Banach spaces of functions that are delta-subharmonic in the unit disc. It begins by analyzing classes of Green type potentials and the generalized fractional integrals introduced by M. M. Djrbashian, which exhibit bounded square integral means. The text utilizes a simplified form of these fractional integrals and explores the properties of the Cauchy type ω-kernel under specific parameter-function choices. Additionally, it examines Blaschke type factors and their transformations, leading to significant inequalities and variable changes that deepen the understanding of these mathematical structures. This comprehensive analysis offers novel insights into the behavior of delta-subharmonic functions and their applications in advanced mathematical research.AI Generated
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AbstractThe present work introduces several Banach spaces of delta-subharmonic in the unit disc of the complex plane functions, the M. M. Djrbashian generalized fractional integrals of which possess bounded square integral means. The union of these spaces coincides with the set of all functions delta-subharmonic in the unit disc. -
Banach Spaces of Functions Delta-Subharmonic in the Half-Plane
Armen Jerbashian, Daniel VargasThe chapter investigates classes of Green type potentials in the upper half-plane, focusing on their fractional integrals with bounded square integral means. It introduces two Blaschke type factors and explores their properties, including holomorphicity and unique zeros. The chapter then delves into the Banach spaces formed by these factors, highlighting their regularizations and subharmonic properties. It provides a detailed analysis of these spaces and their potentials, offering insights into their behavior and applications in complex analysis and functional analysis.AI Generated
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AbstractThe present work gives the introduction and some investigation of several Banach spaces of delta-subharmonic in the half-plane of the complex plane functions, the M.M.Djrbashian generalized fractional integrals of which possess bounded square integral means. -
Weighted Estimates Containing Quasilinear Operators
A. Kalybay, R. OinarovThis chapter delves into the intricate world of weighted estimates, focusing on inequalities that involve quasilinear operators. It begins by defining the interval I and the weight functions u, v, and w, which are non-negative and locally summable. The core of the chapter revolves around the inequalities involving non-negative kernels K(⋅, ⋅) in specific classes. The definitions of these classes are rigorously presented, setting the stage for a deeper exploration of the properties and applications of weighted estimates. The chapter offers a unique perspective on how these estimates behave under different conditions, providing valuable insights for professionals in the field.AI Generated
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AbstractCharacterizations of weighted estimates for a certain class of quasilinear integral operators with kernels are obtained.
- Title
- Operator Theory and Harmonic Analysis
- Editors
-
Alexey N. Karapetyants
Vladislav V. Kravchenko
Elijah Liflyand
Helmuth R. Malonek
- Copyright Year
- 2021
- Publisher
- Springer International Publishing
- Electronic ISBN
- 978-3-030-77493-6
- Print ISBN
- 978-3-030-77492-9
- DOI
- https://doi.org/10.1007/978-3-030-77493-6
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