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Analysis

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Universal interpolation

14. Januar 2013

Andreas Vogt

Seiten: 87-96

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If P n is the polynomial of degree at most n -1 which interpolates a function f :[0,1] → ℝat the nodes 0 ≤ x n 1 < x 2 n < ⋯ < x n n ≤ 1 ( n ∈ ℕ), it is well-known that, even if f is a continuous function, the sequence ( P n ) n ∈ ℕ does not necessarily converge to f . Indeed, for p ∈ [1,∞), there exists an infinitely often differentiable function f and a “nice” system of nodes such that to every measurable function g , there exists a subsequence of ( P n ) n ∈ ℕ that converges in L p to g .
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Generalized Boyd´s indices and applications

14. Januar 2013

Kwok-Pun Ho

Seiten: 97-106

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A generalization of the Boyd indices to Banach function space is introduced. Some applications of these generalized Boyd´s indices to function spaces such as Littlewood–Paley characterization, Fefferman–Stein vector-valued inequalities and wavelet characterization of Banach function space are shown.
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A problem in analysis

14. Januar 2013

Alexander G. Ramm

Seiten: 107-110

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Assume that D ⊂ ℝ 2 is a bounded domain, diffeomorphic to a disc, star-shaped, with a C 1, λ boundary C , λ > 0 , which can be represented in polar coordinates as r = f ( φ ) , where f > 0 is a smooth 2 π -periodic function. Let ψ ± n : = ψ ± n ( φ ): = e ± in φ f n ( φ ). Theorem. Assume that ∫ 0 2 π ψ ± n f 2 ( φ ) d φ = 0 n = 1, 2,⋯ Then f = const .
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New sequences that converge to a generalized Euler constant

14. Januar 2013

Alina Sîntămărian

Seiten: 111-120

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The purpose of the paper is to give some sequences that converge quickly to a generalized Euler constant, i.e. the limit of the sequence (1/ a s +1/( a +1) s +⋯+1/( a + n -1) s -1/(1- s )(( a + n -1) 1- s - a 1- s )) n ∈ ℕ , where a ∈ (0,+∞) and s ∈ (0,1) .
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A uniform version of Laplace´s method for contour integrals

14. Januar 2013

Thorsten Neuschel

Seiten: 121-136

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It is the main objective of this note to establish a “simple and general” version of Laplace´s method for obtaining approximations for complex contour integrals of the type ∫ P e - n p ( t , z ) q ( t , z )d t , as n →∞, holding uniformly with respect to a parameter z ∈ S , where S denotes some compact subset of ℂ . Moreover, two applications of this version are presented for illustration.
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Representation of multipliers on spaces of real analytic functions

14. Januar 2013

Paweł Domański, Michael Langenbruch

Seiten: 137-162

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We consider multipliers on the spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We prove representation theorems in terms of analytic functionals and in terms of holomorphic functions. We characterize Euler differential operators among multipliers. Then we characterize when such operators are surjective or have a continuous linear right inverse on the space of real analytic functions over an interval not containing zero. In particular we solve the problem when Euler differential equation of infinite order has a solution in the space of real analytic functions on an interval not containing zero.
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Infinite pseudo-differential operators on W_M(Rⁿ) space

14. Januar 2013

S. K. Upadhyay, R. N. Yadav, Lokenath Debnath

Seiten: 163-178

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The infinite pseudo-differential operator on W M ( R n ) space is introduced and its various properties are studied. A general class of symbols θ ( x , ξ ) is introduced and then it is proved that the pseudo-differential operator A θ φ is a continuous linear mapping from W M ( R n ) into itself. An L p ( R n ) -boundedness result for the pseudo-differential operator associated with a general class of symbols σ ( x , ξ ) for ξ = u + it is obtained. It is shown that the pseudo-differential operator is a bounded linear operator from L p ( R n ) into L p ( R n ) for 1 < p < ∞ . The Sobolev space of type G s,p ( R n ) is introduced and its properties are studied.

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Erratum to: Approximation schemes for solving disturbed control problems with non-terminal time and state constraints

14. Januar 2013

Nikolay Botkin, Karl-Heinz Hoffmann, Natalie Mayer, Varvara Turova

Seiten: 179-179

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