Harmonic Analysis, Generalized Wavelets and the Generalized Continuous Wavelet Transform on Chébli-Trimèche Hypergroups

An important example of hypergroups is the class of hypergroups X = ℝ ₊ = [0,+ ∞[ defined as follows : Let A be a function defined on X satisfying certain regularity and convexity conditions. Then there exists a unique hypergroup structure on X such that the convolution δ_r * δ_s of the two points masses δ_r and δ_s, with r, s ∈ X, satisfies the equation ∂ ∂ r [ A ( r ) A ( s ) ∂ ∂ r { ∫ X f ( t ) d ( δ r *   δ s ) ( t ) } ] = ∂ ∂ s [ A ( r ) A ( s ) ∂ ∂ r { ∫ X f ( t ) d ( δ r *   δ s ) ( t ) } ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203753712/177c79e4-b1e6-4968-b8f2-68ff8d714780/content/eq1655.tif"/>