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Absolute Convergence of the Double Fourier-Franklin Series
Siberian Mathematical Journal ( IF 0.7 ) Pub Date : 2020-05-01 , DOI: 10.1134/s0037446620030039
G. G. Gevorkyan , M. G. Grigoryan

We prove that, for every 0 < ϵ < 1, there exists a measurable set E ⊂ T = [0, 1] 2 with measure ∣ E ∣ > 1 − ϵ such that, for all f ∈ L 1 ( T ) and 0 < η < 1, we can find $$\tilde f \in {L^1}(T)$$ f ˜ ∈ L 1 ( T ) with $$\int\!\!\!\int_T {{\rm{|}}f(x,y) - \tilde f(x,y){\rm{|}}dxdy \le \eta } $$ ∬ T | f ( x , y ) − f ˜ ( x , y ) | d x d y ≤ η coinciding with f ( x, y ) on E whose double Fourier-Franklin series converges absolutely to f almost everywhere on T .

中文翻译:

双傅里叶-富兰克林级数的绝对收敛

我们证明,对于每一个 0 < ϵ < 1,存在一个可测集 E ⊂ T = [0, 1] 2 其测度 ∣ E ∣ > 1 − ϵ 使得,对于所有 f ∈ L 1 ( T ) 和 0 < η < 1,我们可以找到 $$\tilde f \in {L^1}(T)$$ f ˜ ∈ L 1 ( T ) 与 $$\int\!\!\!\int_T {{\rm {|}}f(x,y) - \tilde f(x,y){\rm{|}}dxdy \le \eta } $$ ∬ T | f ( x , y ) − f ˜ ( x , y ) | dxdy ≤ η 与 E 上的 f ( x, y ) 重合,其双傅立叶-富兰克林级数在 T 上几乎处处绝对收敛到 f。
更新日期:2020-05-01
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