Modulation spaces Mp,q for 0<p, q≦∞

Abstract

The purpose of this paper is to construct modulation spaces Mp,q(Rd) for general 0<p, q≦∞, which coincide with the usual modulation spaces when 1≦p,q≦∞, and study their basic properties including their completeness. Given any g∈S(Rd) such that supp ĝ⊂  {ξ∣|ξ|≦1} and ∑k∈Zdĝ(ξ-αk)≡1, our modulation space consists of all tempered distributions f such that the (quasi)-norm ‖f‖M[g]p,q:≔(∫Rd(∫Rd|f*(Mωg)(x)|pdx)qpdω)1q is finite.