Wavelet Bases for Confinements of the Infrared Divergence

Abstract

We propose the construction of wavelet bases with pseudo-polynomials adapted to the homogeneous Sobolev spaces \(\dot{H}^{s}(\mathbb{R}^{n})\) , s−n/2∈ℕ. They provide a confinement of the infrared divergence by decomposing \(\dot {H}^{s}(\mathbb{R}^{n})\) as a direct sum X⊕Y where X is a “small” space which carries the divergence and Y can be embedded in \(\mathcal{S}'(\mathbb{R}^{n})\) . In the case of \(\dot{H}^{1}(\mathbb{R}^{2})\) we also construct such an orthonormal basis, which provides a confinement of the Mumford process.