Cantor-type uniqueness of multiple trigonometric integrals
نویسندگان
چکیده
منابع مشابه
Uniqueness Questions for Multiple Trigonometric Series
We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges to zero and the other about the series which converge to an integrable function, are asked for four modes of convergence: unrestricted rectangular convergence, spherical convergence, square convergence, and restricted rectangular convergence. We will ...
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In 1870 Cantor proved that representation of a function of one variable by a trigonometric series can be done in only one way. In 1996 Bourgain proved the same thing for spherical convergence and multiple trigonometric series. His proof involves injecting a lot of new ideas into the theory of uniqueness. We give here an exposition of Bourgain’s proof, specialized to the case of dimension 2. Our...
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The issue is uniqueness of representation by multiple trigonometric series. Two basic uniqueness questions, one about series which converge to zero and the other about series which converge to an integrable function, are asked for each of four modes of convergence: unrestricted rectangular convergence, spherical convergence, square convergence, and restricted rectangular convergence. Thus there...
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A subset E of the d-dimensional torus Td is called a set of uniqueness, or U -set, if every multiple trigonometric series spherically converging to 0 outside E vanishes identically. We show that all countable sets are U -sets and also that HJ sets are U -sets for every J . In particular, C × Td−1, where C is the Cantor set, is an H1 set and hence a U -set. We will say that E is a UA-set if ever...
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In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let E be a subset of positive (Lebesgue) measure of the k dimensional torus. The principal result is that the convergence of a trigonometric series on E forces the boundedness of the partial sums alm...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1955
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1955.5.607