A Novel Schwartz Space for the (k,(2)/(n))-Generalized Fourier Transform

2025/07/05 by Faustino, Nelson, Negzaoui, Selma
#22E60 #33C52 #42B10 #46E10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.04064

Abstract

The Schwartz space is the natural ambient space for dealing with the classical Fourier transform beyond L²-spaces. However, when studying more general transforms, such as the (k,a)-generalized Fourier transform, these spaces lose several important properties, such as invariance. To address this issue, we introduce a Schwartz-type space, denoted by S_k,n(ℝ), which is tailored for the (k,(2)/(n))-generalized Fourier transform. This space retains the invariance property of the classical Fourier transform and is useful in cases where the standard Schwartz space is not. To prove afterwards that S_k,n(ℝ) is dense in L^p(dμ_k,n), for 1 ≤ p < ∞, we identify the underlying subspace of functions with bounded support, denoted by D_k,n(ℝ). Finally, we discuss a conjecture regarding extensions to higher dimensions.

Citations

• On the kernel of the (κ,a)-generalized Fourier transform
• Dunkl operators: Theory and applications
• A positive radial product formula for the Dunkl kernel
• Positivity of Dunkl's intertwining operator
• Differential-difference operators associated to reflection groups

Related

• A Characterisation of the Euclidean Fourier transform on the Schwartz space
• On the kernel of the (κ,a)-generalized Fourier transform
• On the Fourier transform of Schwartz functions on Riemannian Symmetric Spaces
• The ρ-Fourier transform
• On the Schwartz space isomorphism theorem for the Riemannian symmetric spaces