Uniform estimates of the coconvex approximation of functions by polynomials

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

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Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

Advances in Difference Equations ◽

10.1186/s13662-020-03124-8 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Abhishek Mishra ◽

Vishnu Narayan Mishra ◽

M. Mursaleen

Keyword(s):

Double Fourier Series ◽

Degree Of Approximation ◽

Trigonometric Approximation ◽

Lipschitz Class ◽

Gamma Delta ◽

Approximation Of Functions ◽

Matrix Summability ◽

Alpha Beta ◽

Generalized Lipschitz Class ◽

Hausdorff Matrix

AbstractIn this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

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DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000045 ◽

2021 ◽

pp. 1-29

Author(s):

ALEXANDER BRUDNYI

Keyword(s):

Disjoint Union ◽

A Priori ◽

Holomorphic Functions ◽

Stable Rank ◽

Open Unit ◽

Uniform Estimates ◽

Approximation Theorems ◽

Type Approximation ◽

Similar Approximation ◽

Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

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Asymptotic Behaviour of the Error of Polynomial Approximation of Functions Like $$\vert x\vert ^{{\alpha }+i{\beta }}$$

Computational Methods and Function Theory ◽

10.1007/s40315-021-00364-x ◽

2021 ◽

Author(s):

Michael I. Ganzburg

Keyword(s):

Asymptotic Behaviour ◽

Polynomial Approximation ◽

Approximation Of Functions ◽

Polynomial Approximation Of Functions

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Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-058-1 ◽

1992 ◽

Vol 35 (4) ◽

pp. 439-448 ◽

Cited By ~ 3

Author(s):

Gerald Beer

Keyword(s):

Approximation Theory ◽

Metric Space ◽

Polish Space ◽

Complete Metric Space ◽

Approximation Of Functions ◽

Lower Envelopes ◽

Constructive Approximation ◽

Hausdorff Approximation ◽

Set Of Points ◽

Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

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