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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

journal of function spaces and applications/journal of function spaces and applications

Modulation spaces<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msup></mml:math>for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi><mml:mo>≦</mml:mo><mml:mi>∞</mml:mi></mml:math>

Modulation spacesMp,qfor0<p,q≦∞