Pseudo-differential operators associated with the Jacobi differential operator and Fourier-cosine wavelet transform

Abstract

Using the inverse of Fourier–Jacobi transform a symbol is defined, and the pseudo-differential operator (p.d.o.) 𝒫α, β (x,D) associated with Jacobi-differential operator in terms of this symbol is defined. It is shown that the p.d.o. is bounded in a certain Sobolev type space associated with the Fourier–Jacobi transform. Continuous Jacobi wavelet transform (JWT) and Fourier-cosine wavelet transform are defined and a reconstruction formula is obtained for Fourier-cosine wavelet transform. Properties of Fourier-cosine wavelet transform are investigated.

Pseudo-differential operators involving Fractional Fourier cosine (sine) transform

“…If we replace the symbol A(x, y) by more general symbol a(x, y), which is no longer a polynomial in y necessarily , we get the pseudo-differential operator A c a,θ defined below. For p.d.o., involving Fourier transform, Hankel transform, fractional Fourier transform, Fourier-Jacobi transform and a singular differential operator, we may refer respectively [11,13], [6], [7,9], [10,12] and [2].…”

Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform

“…We may also refer Pathak [12] who has studied wavelet transform in various function spaces associated with classical Fourier transform. Further, construction of wavelet transforms by using various kind of integral transforms have been carried out by the authors of the field [11,13,17,18,20,30,31]. In [21,25], Prasad et al have constructed and studied key properties of wavelet transform associated with index transforms like Kontorovich-Lebedev transform (KL-transform) and Mehler-Fock transform.…”