Advanced Topics in Mathematical Analysis最新文献_第2页

Composite Submeasures and Supermeasures with Applications to Classical Inequalities 复合子测度和超测度及其在经典不等式中的应用

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-6

S. Dragomir

引用次数: 0

Generalized Double Statistical Weighted Summability and Its Application to Korovkin Type Approximation Theorem 广义双统计加权可和性及其在Korovkin型近似定理中的应用

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-7

U. Kadak, H. Dutta

引用次数: 3

The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions 拓扑代数理论与全纯函数代数的相互作用

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-17

A. Saracco

{"title":"The Interplay Between Topological Algebras Theory and Algebras of Holomorphic Functions","authors":"A. Saracco","doi":"10.1201/9781351142120-17","DOIUrl":"https://doi.org/10.1201/9781351142120-17","url":null,"abstract":"There are several interactions between the general theory of commutative topological algebras over C with unity and that of the holomorphic functions of one or several complex variables. This chapter is intended to be a review chapter of classical results in this subjects and a list of very difficult and long-standing open problems. A big reference section will provide the reader material to dwell into the proofs of theorems and understand more deeply the subjects. This chapter will be divided in two parts. In the first we will review known results on the general theory of Banach algebras, Frechet algebras, LB and LF algebras, as well as the open problems in the field. In the second part we will examine concrete algebras of holomorphic functions, again reviewing known results and open problems. We will mainly focus on the interplay between the two theories, showing how each one can be considered either an object of study per se or a mean to understand better the other subject. 1.1 Topological algebras A topological algebra A over a topological field K is a topological space endowed with continuous operations + : A× A→ A , · : K× A→ A , ? : A× A→ A , such that (A,+, ·) is a vector space over K and (A,+, ?) is an algebra. If the inner product ? is commutative, we will call A a commutative (topological) algebra. We will usually omit the adjective topological in the sequel. A is said commutative algebra with unit if there is 1 ∈ A such that 1 ? a = a ? 1 = a for each a ∈ A. Topological algebras were introduced in 1931 by David van Dantzig in his doctoral theses [47] and they were extensively studied by Izrail Gelfand, Mark Năımark and Georgi Šilov starting from the Forties. Since we are interested in algebras of holomorphic functions, usually K = C, with the Euclidean topology. Moreover we will usually denote a ? b simply by ab. 1.1.1 Banach algebras A commutative algebra B over C with unit 1 is said to be a Banach algebra if it is a Banach space with a norm ‖ · ‖ such that ‖ 1 ‖ = 1, ‖ xy ‖ ≤ ‖ x ‖ ‖ y ‖ , for all x, y ∈ B. Algebras of continuous C-valued bounded functions on a set D, containing constant functions, endowed with the supremum norm, are Banach algebras. Banach algebras of C-valued functions are such that for all elements b of the algebra ‖ b ‖ = ‖ b ‖ . (1.1) A Banach algebra B satisfying the above condition is called an uniform algebra. Actually this condition is very restrictive and the only uniform algebras are algebras of C-valued functions. From now on, B will always denote a Banach algebra. The spectrum of an element b ∈ B is the subset of C σ(b) = {λ ∈ C : λ · 1− is not invertible} For every b ∈ B, the spectrum σ(b) is a non empty compact subset of C. This implies that the only Banach field (up to an isometric isomorphism) is C (GelfandMazur theorem). 1.1.1.1 Banach algebras: characters and spectrum A character of B is a homomorphism of algebras χ : B → C. Every character χ of a Banach algebra is continuous. In particular the no","PeriodicalId":319080,"journal":{"name":"Advanced Topics in Mathematical Analysis","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130924356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On Difference Operators and Their Applications 差分算子及其应用

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-12

P. Baliarsingh, H. Dutta

引用次数: 1

Birkhoff-James Orthogonality and Its Application in the Study of Geometry of Banach Space Birkhoff-James正交及其在巴拿赫空间几何研究中的应用

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-8

K. Paul, D. Sain

引用次数: 2

Some Subclasses of Analytic Functions and Their Properties 解析函数的几个子类及其性质

Advanced Topics in Mathematical Analysis Pub Date : 2019-01-08 DOI: 10.1201/9781351142120-10

N. Mustafa, Veysel Nezir, H. Dutta

引用次数: 0

Composite Ostrowski and Trapezoid Type Inequalities for Riemann-Liouville Fractional Integrals of Functions with Bounded Variation 有界变分函数的Riemann-Liouville分数积分的复合Ostrowski和梯形不等式

Advanced Topics in Mathematical Analysis Pub Date : 1900-01-01 DOI: 10.1201/9781351142120-13

S. Dragomir

引用次数: 2