Pointwise convergence of \(n\)-dimensional Hermite expansions (Q1916827)

The authors prove the following theorem. Suppose that an integrable function \(f(x)\), \(x\in\mathbb{R}^n\), is piecewise smooth with respect to \(x=0\). Then the spherical partial sums of the \(n\)-dimensional Fourier-Hermite series of \(f\) converge if and only if the smoothness index (introduced in this paper) satisfies the inequality \(j(f; 0)\geq[(n- 3)/2]\). The authors also prove the following auxiliary result, interesting in itself. Let \(a\neq 0\), then the series \(\sum^\infty_{k=1} k^{-\beta}e^{ia\sqrt k}\) converges if and only if \(\beta> 1/2\).