On function spaces with fractional Fourier transform in weighted Lebesgue spaces

Abstract

Let w and co be weight functions on R-d. In this work, we define A(alpha,p)(w,omega)(R-d) to be the vector space of f is an element of L-w(1) (R-d) such that the fractional Fourier transform F(alpha)f belongs to L-omega(p)(R-d) for 1 <= p < infinity. We endow this space with the sum norm parallel to f parallel to A(alpha,p)(w,omega) = parallel to f parallel to(1,w) + parallel to F(alpha)f parallel to(pw) and show that A(alpha,p)(w,omega)(R-d) becomes a Banach space and invariant under time-frequency shifts. Further we show that the mapping y -> T(y)f is continuous from R-d into A(alpha,p)(w,omega)(R-d) the mapping z -> M(z)f is continuous from R-d into A(alpha,p)(w,omega)(R-d) and A(alpha,p)(w,omega)(R-d) is a Banach module over L-w(1)(R-d) with Theta convolution operation. At the end of this work, we discuss inclusion properties of these spaces.