اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLogarithmic Potential Theory with Applications to Approximation Theory
223
0
0.0
(
0
)
تأليف
E.B. Saff
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to explore the notions of Fekete points, logarithmic capacity, and Chebyshev constant through a variety of examples and exercises. Many of the fundamental theorems of potential theory, such as Frostmans theorem, the Riesz Decomposition Theorem, the Principle of Domination, etc., are given along with essential ideas for their proofs. Equilibrium measures and potentials and their connections with Green functions and conformal mappings are presented. Moreover, we discuss extensions of the classical potential theoretic results to the case when an external field is present.
قيم البحث
اقرأ أيضاً
In the present paper we develop the theory of minimization for energies with multivariate kernels, i.e. energies, in which pairwise interactions are replaced by interactions between triples or, more generally, $n$-tuples of particles. Such objects, w
hich arise naturally in various fields, present subtle differences and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, explore rotationally invariant energies on the sphere, and present a variety of interesting examples, in particular, some optimization problems in probabilistic geometry which are related to multivaria
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm of a posit
ive random variable (or the logarithm of a sum of positive random variables). The integral representation of the logarithm is proved useful in a variety of information-theoretic applications, including universal lossless data compression, entropy and differential entropy evaluations, and the calculation of the ergodic capacity of the single-input, multiple-output (SIMO) Gaussian channel with random parameters (known to both transmitter and receiver). This integral representation and its variants are anticipated to serve as a useful tool in additional applications, as a rigorous alternative to the popular (but non-rigorous) replica method (at least in some situations).
We give applications of Foliation Theory to the Classical Invariant Theory of real orthogonal representations, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is
a generating set. And the introduction of a class of generalized polarizations which, in the case of representations of finite groups, always generates the algebra of invariants of their diagonal representations.
In this paper the double-sided Taylors approximations are studied. A short proof of a well-known theorem on the double-sided Taylors approximations is introduced. Also, two new theorems are proved regarding the monotonicity of such approximations. Th
en we present some new applications of the double-sided Taylors approximations in the theory of analytic inequalities.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
