Uniqueness for spherically convergent multiple trigonometric series
نویسنده
چکیده
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 exposition includes a fairly general method for finding maximal elements without resorting to the Axiom of Choice. 1 Background The first major question that arose in the history of Fourier series was this. Determine which functions mapping the interval [0, 2π) = T into the complex numbers can be represented in the form
منابع مشابه
Some spherical uniqueness theorems for multiple trigonometric series
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 con...
متن کاملSets of Uniqueness for Spherically Convergent Multiple Trigonometric Series
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...
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 کاملUniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums
If at each point of a set of positive Lebesgue measure, every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier ser...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2000