A direct application of Zorn's Lemma gives that every lipschitz map f : X subset of Q(p)(n) -> Q(p)(l) has an extension to a lipschitz map (f) over tilde : Q(p)(n) -> Q(p)(l). This is analogous, but more eas...
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A direct application of Zorn's Lemma gives that every lipschitz map f : X subset of Q(p)(n) -> Q(p)(l) has an extension to a lipschitz map (f) over tilde : Q(p)(n) -> Q(p)(l). This is analogous, but more easy, to Kirszbraun's Theorem about the existence of lipschitz extensions of lipschitz maps S subset of R-n -> R-l. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's Theorem. In the present paper, we prove in the p-adic context that (f) over tilde can be taken definable when f is definable, where definable means semi-algebraic or subanalytic (or, some intermediary notion). We proceed by proving the existence of definable, lipschitz retractions of Q(p)(n) to the topological closure of X when X is definable.
We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague converg...
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We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology. (C) 2019 Elsevier B.V. All rights reserved.
We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally lipschitzcontinuous with some constant C is piecewise (globally on each piece) lipschitzcontinuous with possibly so...
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We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally lipschitzcontinuous with some constant C is piecewise (globally on each piece) lipschitzcontinuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f(y) : X(y) subset of Q(p)(n) -> Q(p) depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q(p). These results are p-adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p-adic fields, we need to introduce new methods adapted to the p-adic situation.
In this paper, we introduce a new type lambda-Bernstein operators with parameter lambda epsilon[- 1, 1], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergenc...
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In this paper, we introduce a new type lambda-Bernstein operators with parameter lambda epsilon[- 1, 1], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of B-n,B-lambda(f;x) to f (x), and we see that in some cases the errors are smaller than B-n(f) to f.
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