اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدVector-valued Fourier hyperfunctions and boundary values
115
0
0.0
(
0
)
تأليف
Karsten Kruse
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This work is dedicated to the development of the theory of Fourier hyperfunctions in one variable with values in a complex non-necessarily metrisable locally convex Hausdorff space $E$. Moreover, necessary and sufficient conditions are described such that a reasonable theory of $E$-valued Fourier hyperfunctions exists. In particular, if $E$ is an ultrabornological PLS-space, such a theory is possible if and only if E satisfies the so-called property $(PA)$. Furthermore, many examples of such spaces having $(PA)$ resp. not having $(PA)$ are provided. We also prove that the vector-valued Fourier hyperfunctions can be realized as the sheaf generated by equivalence classes of certain compactly supported $E$-valued functionals and interpreted as boundary values of slowly increasing holomorphic functions.
قيم البحث
اقرأ أيضاً
We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunct
ion, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, Baumer, Lumer and Neubrander and Langenbruch.
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $fcolon Gto E$. A function $fin C(G,E)$ is a generalized polynomial, if there is an $nge 0$ such that $Delta_{h_1
} ldots Delta_{h_{n+1}} f=0$ for every $h_1 ,ldots , h_{n+1} in G$, where $Delta_h$ is the difference operator. We say that $fin C(G,E)$ is a polynomial, if it is a generalized polynomial, and the linear span of its translates is of finite dimension; $f$ is a w-polynomial, if $ucirc f$ is a polynomial for every $uin E^*$, and $f$ is a local polynomial, if it is a polynomial on every finitely generated subsemigroup. We show that each of the classes of polynomials, w-polynomials, generalized polynomials, local polynomials is contained in the next class. If $G$ is an Abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. We introduce the classes of exponential polynomials and w-expo-nential polynomials as well, establish their representations and connection with polynomials and w-polynomials. We also investigate spectral synthesis and analysis in the class $C(G,E)$. It is known that if $G$ is a compact Abelian group and $E$ is a Banach space, then spectral synthesis holds in $C(G,E)$. On the other hand, we show that if $G$ is an infinite and discrete Abelian group and $E$ is a Banach space of infinite dimension, then even spectral analysis fails in $C(G,E)$. If, however, $G$ is discrete, has finite torsion free rank and if $E$ is a Banach space of finite dimension, then spectral synthesis holds in $C(G,E)$.
We prove that for every Banach space $Y$, the Besov spaces of functions from the $n$-dimensional Euclidean space to $Y$ agree with suitable local approximation spaces with equivalent norms. In addition, we prove that the Sobolev spaces of type $q$ ar
e continuously embedded in the Besov spaces of the same type if and only if $Y$ has martingale cotype $q$. We interpret this as an extension of earlier results of Xu (1998), and Martinez, Torrea and Xu (2006). These two results combined give the characterization that $Y$ admits an equivalent norm with modulus of convexity of power type $q$ if and only if weakly differentiable functions have good local approximations with polynomials.
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k
times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach sp
ace $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_infty(mu,X)$ and $L_1(mu,X)$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
