Uniform approximation by spherical Fourier sums on classes of functions defined by polyharmonic operators (Q1812668)

Let \({\mathcal L}^ r_ \infty(Q_ N)\) denote the class of \(2\pi\)-periodic functions \(f\) of the form \[ f(x)=(2\pi)^{-N}\int_{Q_ N}\varphi(x- u)\sum_{| m|\neq 0}| m|^{-r}e^{imu}du,\;r>0, \] \((\int_{Q_ N}\varphi dx=0\) and \(|\varphi(x)|\leq 1\) a.e., \(Q_ N=\{x:x\in E_ N,| x_ j|\leq\pi,\;j=1,\dots,N\})\). The author proved the following asymptotic result: The quantity \[ \varepsilon_ R=\sup_{f\in{\mathcal L}_ \infty^ r(Q_ N)}\| f- S^ 0_ R(f)\|_ C \] satisfies the equality \[ \varepsilon_ R=R^{-r}L^ N_ R+O(R^{(N-3)/2-r}),\quad R\to\infty, \] where \(L^ N_ R=(2\pi)^{-N}\int_{Q_ N}\left|\sum_{| m|<R}e^{imx}\right| dx\) are the Lebesgue constants of the spherical Fourier sums \(S^ 0_ R(f)\).