Banach function norms via Cauchy polynomials and applications

2015 ◽  
Vol 26 (10) ◽  
pp. 1550083 ◽  
Author(s):  
Giuseppina Anatriello ◽  
Alberto Fiorenza ◽  
Giovanni Vincenzi

Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial [Formula: see text]. We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.


1997 ◽  
Vol 1 (1) ◽  
pp. 52-65
Author(s):  
Hye-Joon Yoon

Area studies, as a newly fashionable field of academic research, needs to recognize its less likely precedents if it is going to secure for itself a fresh start. The question of “desire” is relevant here because it indicates the less value-free aspects in its genealogy. As shown in Emma Bovary's embellished representation of Paris at her provincial home, an understanding of an area often reflects the particular needs and desires of the one who understands that area. Such restricted and restricting views of an area repeats itself outside the world of literary fictions, as is shown by the example of Guizot's picture of Europe in which his own country is given a privileged place as the very center of Western civilization itself. An instructive case showing the thin line between the projected desire of one who strives to know a geographical area and the scientific purity of the labor itself is further offered by Napoleon Bonaparte's heavy reliance on Orientalist scholarship in his invasion of Egypt. Moving further east from Egypt to China, we witness the denigrating remarks on China made by the great German thinkers of the past century, Hegel and Weber. Although their characterization of Chinese culture could find echoes in unbiased empirical research, they reveal all the same the trace of Europeans' desire to affirm their superiority over the supposedly inferior and false civilization of the East. Similarly, the Americans who divided the Korean peninsular at the 38th Parallel, with unquestioning confidence in their knowledge of the area and in the justice of their action, rightfully deserve their place in the tradition of Western area studies of serving the needs to dominate, control and exploit an objectified overseas territory. He assumed that words had kept their meaning, that desires still pointed in a single direction, and that ideas retained their logic; and he ignored the fact that the world of speech and desires has known invasions, struggles, plundering, disguises, ploys. From these elements, however, genealogy retrieves an indispensable restraint: it must record the singularity of events outside of any monotonous finality; it must seek them in the most unpromising places, in what we tend to feel is without history—in sentiments, love, conscience, instincts; it must be sensitive to their recurrence, not in order to trace the gradual curve of their evolution, but to isolate the different scenes where they engaged in different roles. — Michel Foucault, “Nietzsche, Genealogy, History” (Foucault 139–40).



2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song


2008 ◽  
Vol 28 (5) ◽  
pp. 1369-1375 ◽  
Author(s):  
ERIC BEDFORD ◽  
KYOUNGHEE KIM

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.



2003 ◽  
Vol 11 (2) ◽  
pp. 171-181 ◽  
Author(s):  
WAN JUNREN

The present state of the study of philosophy in China cannot be understood if the introduction of Western philosophy, including Marxism, in the early decades of the 20th century is ignored. During the first half of the past century, philosophy flourished at Peking University with a heavy emphasis on social and political theory, as well as at Tsinghua University, which focused more on professionalization of the discipline, logic, epistemology and the history of philosophy. From 1957 to 1979, philosophical studies suffered severely under an increasingly unfavourable political climate. It is only recently that departments of philosophy have been reconstructed, not only in Beijing but also in all major universities. Like their colleagues in other disciplines, many philosophers are nowadays struggling with the dichotomy between the old and the new, the Chinese and the Western tradition, and resisting an uncritical absorption of Western ideas. Finally, some observations are made on the lectures that Jürgen Habermas gave in China in 2001.



2007 ◽  
Vol 49 (3) ◽  
pp. 431-447 ◽  
Author(s):  
MASATO KIKUCHI

AbstractLet X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.



2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ruimin Wu ◽  
Songbai Wang

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .



The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H 2 μ ( k, λ ) = -d 2 / d x 2 + kx 2 + λx 2 μ , ( k, λ ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ . The eigenvalues, as functions of λ , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.



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