Localization For Fourier Series Research Articles

Necessary conditions for the weak generalized localization of Fourier series with “lacunary sequence of partial sums”

It has been established that, on the subsets \( \mathbb{T}^N = [ - \pi ,\pi ]^N \) describing a cross W composed of N-dimensional blocks, \( W_{x_s x_t } = \Omega _{x_s x_t } \times [ - \pi ,\pi ]^{N - 2} (\Omega _{x_s x_t } \) is an open subset of [−π, π]2) in the classes \( L_p (\mathbb{T}^N ),p > 1 \), a weak generalized localization holds, for N ≥ 3, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums \( S_n (x;f)(x \in \mathbb{T}^N ,f \in L_p ) \) of these series corresponds the number n = (n1,…, nN) ∈ ℤ+N, some components nj of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating W) of the N-dimensional blocks \( W_{x_s x_t } \) as well as of the structure and geometry of \( W_{x_s x_t } \). In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block.

On pointwise convergence and localization for Fourier series on compact Lie groups

On pointwise convergence and localization for Fourier series on compact Lie groups

Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes Lp

Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes Lp

Localization for Fourier Series on SU(2)

Localization for Fourier Series on SU(2)

Localization for Fourier series on 𝑆𝑈(2)

Let N(b) be the space of functionsf E L1(SU(2)) which vanish on a neighborhood Vf of b. The main results of this paper are Theorems A-C below. THEOREM A. Let b E SU(2), fE N(b). If all of the first derivatives off (in the distribution sense) are functions in LP(SU(2)) for some p > 3/2, then limn_ , Sn(f: b) = 0. If p < 3/2, there is a function f E N(b) whose first derivatives are all functions in LP(SU(2)) and such that limn_.o Sn(f: b) does not exist. THEOREM B. Iff fE N(b) n N(b) and the first derivatives of f are functions in L1(SU(2)), then limn_ c, Sn(f b)=0. Say a functionf E L1(SU(2)) is of bounded variation if its first derivatives are all measures.

The generalized treatment of the principle of localization for fourier series in fundamental systems of functions

The generalized treatment of the principle of localization for fourier series in fundamental systems of functions