Interpolation Theorem Research Articles

A note on an invariant distance of the bidisk

In this short paper, we discuss relation between an invariant distance of the bidisk and Kreĭn space geometry. In particular, an interpolation theorem for rational maps with respect to our invariant distance is proven.

On Haagerup noncommutative quasi [formula omitted] spaces

Let M be a σ-finite von Neumann algebra, equipped with a normal faithful state φ, and let A be a maximal subdiagonal subalgebra of M. We have proved that for 0<p<1, Hp(A) is independent of φ. Furthermore, in the case that A is a type 1 subdiagonal subalgebra, we have obtained an interpolation theorem for Hp(A) in the case where 0<θ<1, 0<p0,p1≤∞ and p=1−θp0+θp1≥1.

Broadcast signcryption scheme with equality test in smart transportation system

With the generation of massive traffic information in the smart transportation system, the traffic control center efficiently utilizes broadcast communication to send multiple messages to multiple vehicles. Besides, diversified privacy disclosure and security attack issues also emerged spontaneously. To achieve secure communication between the traffic control center and vehicles in the smart transportation system, we design a broadcast signcryption scheme with equality test in the smart transportation system based on the certificateless cryptosystem and equality test. The scheme realizes message confidentiality and vehicle privacy by using the Lagrange interpolation theorem to encrypt messages and vehicle identities, while also achieving classify ciphertext by using the equality test and facilitate road traffic information management. Through numerical experiment analysis, the proposed work has higher operation efficiency and is more suitable for application in smart transportation systems.

Stević–Sharma‐type operators between Bergman spaces induced by doubling weights

Using Khinchin's inequality, Geršgorin's theorem, and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević–Sharma‐type operators between weighted Bergman spaces and and the sum of weighted differentiation composition operators with different symbols from the weighted Bergman spaces to . The estimates of those between Bergman spaces remove all the restrictions of a result of Stevic, Sharma, and Bhat. As a by‐product, we also get an interpolation theorem for Bergman spaces induced by doubling weights.

Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure

We prove that every open Riemann surface [Formula: see text] is the complex structure of a complete surface of constant mean curvature [Formula: see text] ([Formula: see text]) in the three-dimensional hyperbolic space [Formula: see text]. We go further and establish a jet interpolation theorem for complete conformal [Formula: see text] immersions [Formula: see text]. As a consequence, we show the existence of complete densely immersed [Formula: see text] surfaces in [Formula: see text] with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in [Formula: see text] which is also established in this paper.

Nonassociative Lp-spaces and embeddings in noncommutative Lp-spaces

We define a notion of nonassociative Lp-space associated to a JBW⁎-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ. In the particular case of JW⁎-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp-spaces. We also make the link with the nonassociative Lp-spaces of Iochum associated to JBW-algebras and the investigation of contractively complemented subspaces of noncommutative Lp-spaces. More precisely, we show that our nonassociative Lp-spaces contain isometrically the Lp-spaces of Iochum and that all tracial nonassociative Lp-spaces from JW⁎-factors arise as positively contractively complemented subspaces of noncommutative Lp-spaces.

Marcinkiewicz interpolation theorem for Hardy type spaces and its applications

В статье приводится ряд утверждений, подобных теореме Марцинкевича об интерполировании операторов. Отличие от классических форм этой теоремы состоит в том, что пространства суммируемых функций заменяются на некоторые классы функций, являющиеся расширениями различных пространств Харди. Указаны также некоторые приложения этих результатов: к обобщению теоремы вложения Карлесона и неравенств Харди-Литтлвуда для аналитических функций из классов Харди. Библиография: 41 название.

Marcinkiewicz's interpolation theorem for Hardy-type spaces and its applications

A series of results similar to Marcinkiewicz's theorem on the interpolation of operators is put forward. The difference from the classical forms of this theorem is that spaces of integrable functions are replaced by some function classes that are extensions of various Hardy spaces. Some applications of these results to the extension of Carleson's embedding theorem and the Hardy-Littlewood inequalities for analytic functions in Hardy classes are presented. Bibliography: 41 titles.

Inductive methods for Hardy inequality on trees

We study the Hardy inequality on at most countable rooted tree. The main known criteria in the lower - triangle case for this inequality are two Arcozzi - Rochberg - Sawyer criteria and the capacity criterion. In the survey we show that these two criteria are connected with the criteria for the Hardy inequalities for the sequences, for the Hardy inequality on an interval of the real axis and for the trace inequalities with the Riesz potentials. We provide the examples from the literature, when the trace inequality or another statement is characterized in terms of the validity of the Hardy inequality on the tree. We simplify two known proofs of the Arcozzi - Rochberg - Sawyer criteria, which are based on the Marcinkiewicz interpolation theorem and on the capacity criterion. We provide new proofs for Arcozzi - Rochberg - Sawyer criteria, which are based on the induction in the tree, the inductive formula for the capacity and the formula of integration by parts. The latter of the proofs is written for the Hardy inequality on the tree with a boundary and for the Hardy inequality over the family of all binary cubes. In the diagonal case this proof provides an optimal constant $p$, which coincides with the Bennett constant in the Hardy inequality for the sequences. In the general case we provide a few new inductive criteria for the validity of the Hardy inequality in terms of the existence of a family of functions satisfying an inductive relation. One of these criteria is applied in the proof of a theorem containing additional equivalent conditions for the validity of the Hardy inequality on the trees in the diagonal case.

Interpolation theorem for discrete net spaces

Interpolation theorem for discrete net spaces

Bisequent Calculus for Four-Valued Quasi-Relevant Logics: Cut Elimination and Interpolation

We present a uniform syntactical characterisation of the class of quasi-relevant logics which are four-valued extensions of the basic relevant logic B of Meyer and Routley. All these logics are obtained by the addition of suitable quasi-relevant implications to the four-valued logic of First Degree Entailment FDE. So far they were characterised axiomatically and semantically in several ways but did not obtain a special proof-theoretic treatment. To this aim a generalised form of sequent calculus called bisequent calculus (BSC) is applied. In BSC rules operate on the ordered pairs of ordinary sequents. It may be treated as the weakest kind of system in the rich family of generalised sequent calculi operating on items which are some collections of ordinary sequents, like hypersequents or nested sequents. It is shown that all logics under consideration have cut-free characterisation in BSC which satisfies the subformula property and yields decidability. It is also shown that the interpolation theorem holds for these logics if their language is enriched with additional negation.

On duality and model theory for polyadic spaces

This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.

Interpolation in Term Functor Logic

Given some links between Lyndon’s Interpolation Theorem, term distribution, and Sommers and Englebretsen’s logic, in this contribution we attempt to capture a sense of interpolation for Sommers and Englebretsen’s Term Functor Logic. In order to reach this goal we first expound the basics of Term Functor Logic, together with a sense of term distribution, and then we offer a proof of our main contribution.

From sphere packing to Fourier interpolation

Viazovska’s solution of the sphere packing problem in eight dimensions is based on a remarkable construction of certain special functions using modular forms. Great mathematics has consequences far beyond the problems that originally inspired it, and Viazovska’s work is no exception. In this article, we’ll examine how it has led to new interpolation theorems in Fourier analysis, specifically a theorem of Radchenko and Viazovska.

Definite descriptions and hybrid tense logic

We provide a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms. It is formalised by means of a tableau calculus working on sat-formulas. A particular theory of DD exploited here is essentially based on the approach of Russell, but with descriptions treated as genuine terms. However, the reductionist aspect of the Russellian approach is retained in several ways. Moreover, a special form of tense definite descriptions is formally developed. A constructive proof of the interpolation theorem for this calculus is given, which is an extension of the result provided by Blackburn and Marx.

Lieb's Inequality for Continuous Modulated Shearlet Transform on LCA Groups

We have proved a version of Lieb's inequality for continuous modulated shearlet transform on locally compact Abelian (LCA) groups. An alternative proof for the inequality has also been provided using Riesz-Thorin interpolation theorem.

Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.

Privacy-Preserving Fine-Grained Redaction with Policy Fuzzy Matching in Blockchain-Based Mobile Crowdsensing

The redactable blockchain has emerged as a promising technique in mobile crowdsensing, allowing users to break immutability in a controlled manner selectively. Unfortunately, current fine-grained redactable blockchains suffer two significant limitations in terms of security and functionality, which severely impede their application in mobile crowdsensing. For security, the transparency of the blockchain allows anyone to access both the data and policy, which consequently results in a breach of user privacy. Regarding functionality, current solutions cannot support error tolerance during policy matching, thereby limiting their applicability in various situations, such as fingerprint-based and face-based identification scenarios. This paper presents a privacy-preserving fine-grained redactable blockchain with policy fuzzy matching, named PRBFM. PRBFM supports fuzzy policy matching and partitions users’ privileges without compromising user privacy. The idea of PRBFM is to leverage threshold linear secret sharing based on the Lagrange interpolation theorem to distribute the decryption keys and chameleon hash trapdoors. Additionally, we have incorporated a privacy-preserving policy matching delegation mechanism into PRBFM to minimize user overhead. Our security analysis demonstrates that PRBFM can defend against the chosen-ciphertext attack. Moreover, experiments conducted on the FISCO blockchain platform show that PRBFM is at least 7.8 times faster than existing state-of-the-art solutions.

Fractional integrals associated with Radon transforms

Abstract We obtain sharp L p {L^{p}} - L q {L^{q}} estimates for fractional integrals generated by Radon transforms of the following three types: The classical Radon transform over the set of all hyperplanes in ℝ n {\mathbb{R}^{n}} , the Strichartz transversal transform over only those hyperplanes which meet the last coordinate axis, and the Radon transform associated with paraboloids. The method relies on a version of Stein’s interpolation theorem for analytic families of operators communicated by L. Grafakos.

Domains of holomorphy in Stein spaces

We prove an interpolation theorem for domains in Stein spaces that are locally Stein outside rare analytic sets and improve several existing results in this area. This is then applied to the Levi problem in Stein spaces.