کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
9502788 | 1631576 | 2005 | 17 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
The optimal form of selection principles for functions of a real variable
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موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
آنالیز ریاضی
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
Let T be a nonempty set of real numbers, X a metric space with metric d and XT the set of all functions from T into X. If fâXT and n is a positive integer, we set ν(n,f)=supâi=1nd(f(bi),f(ai)), where the supremum is taken over all numbers a1,â¦,an,b1,â¦,bn from T such that a1⩽b1⩽a2⩽b2⩽â¯â©½an⩽bn. The sequence {ν(n,f)}n=1â is called the modulus of variation of f in the sense of Chanturiya. We prove the following pointwise selection principle: If a sequence of functions{fj}j=1ââXTis such that the closure in X of the set{fj(t)}j=1âis compact for eachtâTand(â)limnââ(1nlimâsupjââν(n,fj))=0,then there exists a subsequence of{fj}j=1â, which converges in X pointwise on T to a functionfâXTsatisfyinglimnââν(n,f)/n=0. We show that condition (â) is optimal (the best possible) and that all known pointwise selection theorems follow from this result (including Helly's theorem). Also, we establish several variants of the above theorem for the almost everywhere convergence and weak pointwise convergence when X is a reflexive separable Banach space.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 310, Issue 2, 15 October 2005, Pages 609-625
Journal: Journal of Mathematical Analysis and Applications - Volume 310, Issue 2, 15 October 2005, Pages 609-625
نویسندگان
Vyacheslav V. Chistyakov,