On convex multiplies of convergence of some classes of bivariate Fourier series (Q1061336)

The sequence \(\lambda =\{\lambda (k)\}\), \(k\in {\mathbb{N}}^ 2\), is quasi- convex if \[ \sum^{\infty}_{k_ 1=0}\sum^{\infty}_{k_ 2=0}(k_ 1+1)(k_ 2+1)| \Delta^ 2_ 1\Delta^ 2_ 2\lambda (k)| <\infty \] where \(\Delta^ 2_ 1\) and \(\Delta^ 2_ 2\) denote the second differences in the first and in the second variable, \(\lambda\) is convex by each variable if \(\Delta^ 2_ i\lambda (k)\geq 0\), \(i=1,2\). Let \({\mathfrak M}=\{m(\kappa)\}\) be a monotonically increasing sequence of points in \({\mathbb{N}}^ 2\) and \[ s_ mf(x)=K_ 4\sum^{m_ 1}_{k_ 1=-m_ 1}\sum^{m_ 2}_{k_ 2=-m_ 2}\hat f(k)\exp (i(k,x)) \] (f\(\in L\), i.e. f \(2\pi\)-periodic, integrable). Then \(f\in L_ F({\mathfrak M})\Leftrightarrow\) \(\| s_{m(\kappa)}f- s_{m(\nu)}f\|_ L=o(1)\) (\(\kappa\),\(\nu\) \(\to \infty).\) The author gives a sufficient and necessary condition for a quasi-convex sequence \(\lambda =\{\lambda (k)\}\), convex by each variable with \(\lambda\) (k)\(\to 0\) as \(| k| \to \infty\) to be a multiplier sequence from \(L_ F({\mathfrak N})\) into \(L_ F({\mathfrak M})\), \({\mathfrak N}\) and \({\mathfrak M}\) geometrically close. The condition depends on the multipliers \(\lambda\) (k) either in the vertices of the rectangles of \({\mathfrak M}\), or next to the points where the sides of the rectangles of \({\mathfrak M}\) and \({\mathfrak N}\) intersect each other, and on the distances between parallel sides.