Some spherical uniqueness theorems for multiple trigonometric series
نویسندگان
چکیده
We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function f(x) which is bounded below by an integrable function, then the series is the Fourier series of f(x) if the coefficients of the multiple trigonometric series satisfy a mild growth condition. As a consequence, we show that if a multiple trigonometric series is spherically convergent everywhere to an everywhere finite integrable function f(x), then the series is the Fourier series of f(x). We also show that a singleton is a set of uniqueness. These results are generalizations of a recent theorem of J. Bourgain and some results of V. Shapiro.
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 ...
متن کاملConvergence, Uniqueness, and Summability of Multiple Trigonometric Series
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...
متن کاملUniqueness for Higher Dimensional Trigonometric Series
Five uniqueness questions for multiple trigonometric series are surveyed. If a multiple trigonometric series converges everywhere to zero in the sense of spherical convergence, of unrestricted rectangular convergence, or of iterated convergence, then that series must have every coefficient being zero. But the cases of square convergence and restricted rectangular convergence lead to open questi...
متن کاملA Survey of Uniqueness Questions in Multiple Trigonometric Series
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...
متن کاملNew Uniqueness Theorems for Trigonometric Series
A uniqueness theorem is proved for trigonometric series and another one is proved for multiple trigonometric series. A corollary of the second theorem asserts that there are two subsets of the d-dimensional torus, the first having a countable number of points and the second having 2d points such that whenever a multiple trigonometric series "converges" to zero at each point of the former set an...
متن کامل