On the Set of Points of Convergence of a Lacunary Trigonometric Series and the Equidistribution Properties of Related Sequences
نویسنده
چکیده
It is well known from the classical theory of trigonometric series that the series (1) cannot converge except possibly for values of x in a set of zero Lebesgue measure . Our object is to discover how `thin' this set is for various types of sequence {nk } . The first result of this kind is due to P . Turan who proved, in 1941, that the series (1) converges absolutely in a set of positive logarithmic capacity in the case
منابع مشابه
On Lacunary Statistical Limit and Cluster Points of Sequences of Fuzzy Numbers
For any lacunary sequence $theta = (k_{r})$, we define the concepts of $S_{theta}-$limit point and $S_{theta}-$cluster point of a sequence of fuzzy numbers $X = (X_{k})$. We introduce the new sets $Lambda^{F}_{S_{theta}}(X)$, $Gamma^{F}_{S_{theta}}(X)$ and prove some inclusion relaions between these and the sets $Lambda^{F}_{S}(X)$, $Gamma^{F}_{S}(X)$ introduced in ~cite{Ayt:Slpsfn} by Aytar [...
متن کاملA Class of Convergent Series with Golden Ratio Based on Fibonacci Sequence
In this article, a class of convergent series based on Fibonacci sequence is introduced for which there is a golden ratio (i.e. $frac{1+sqrt 5}{2}),$ with respect to convergence analysis. A class of sequences are at first built using two consecutive numbers of Fibonacci sequence and, therefore, new sequences have been used in order to introduce a new class of series. All properties of the se...
متن کاملOn $I$-statistical and $I$-lacunary statistical convergence of order $alpha$
In this paper, following a very recent and new approach, we further generalize recently introduced summability methods, namely, $I$-statistical convergence and $I$-lacunary statistical convergence (which extend the important summability methods, statistical convergence and lacunary statistical convergence using ideals of $mathbb{N}$) and introduce the notions of $I$-statis...
متن کاملNon-Newtonian Fuzzy numbers and related applications
Although there are many excellent ways presenting the principle of the classical calculus, the novel presentations probably leads most naturally to the development of the non-Newtonian calculus. The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Since this self-contained work is intended for a wide audience, inc...
متن کاملOn permutation-invariance of limit theorems
By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d. sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d. sequenc...
متن کامل