Pointwise Fourier inversion on tori and other compact manifolds
A number of results about pointwise convergence of eigenfunction expansions on compact manifolds are proved. It is necessary to remark among this results Proposition 2 in Section 2 on eigenfunction expansions on a two-dimensional manifold. Using the Sogge inequality the author proved the convergence of the expansions of functions from some Besov spaces (more exactly, from the special kind of them which are known as Nikol'skij spaces). The most interesting results concerned with multiple Fourier series are in Section 5. It is proved in Proposition 6 that the expansion of a piecewise smooth function on \(\mathbb T^3\) with simple jump across some sphere converges at each point except the center of this sphere. If the piecewise function on \(\mathbb T^3\) has simple jump across some two-dimensional manifold which does not contain a sphere, then convergence holds for each point from \(\mathbb T^3\) (Proposition 7). In Section 6 the author shows that for an arbitrary countable set \(S\subset\mathbb T^3\) there exist a piecewise smooth function such that its Fourier expansion converges outside \(S\) and diverges (boundedly) on \(S\).
scientific article; zbMATH DE number 1370063