Fourier expansions of functions with bounded variation of several variables

Abstract:

In the first part of the paper we establish the pointwise convergence as $t\rightarrow +\infty$ for convolution operators $\int _{\mathbb {R}^{d}}t^{d}K\left ( ty\right ) \varphi (x-y)dy$ under the assumptions that $\varphi (y)$ has integrable derivatives up to an order $\alpha$ and that $\left | K(y)\right | \leq c\left ( 1+\left | y\right | \right ) ^{-\beta }$ with $\alpha +\beta >d$. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.