Calderón’s reproducing formula and uncertainty principle for the continuous wavelet transform associated with the q-Bessel operator

A variation of the Lq,p‐uncertainty inequalities of Heisenberg‐type for symmetric q‐integral transforms

“…Various mathematical aspects of the uncertainty principle are studied; theorems of Hardy, Heisenberg-Pauli-Weyl type uncertainty inequalities and so forth are established for these transformations; see previous works. 8,15,17,20,24,25,34,35 In this section, we present new uncertainty inequalities of Heisenberg-type for the q-Fourier-cosine transform, 11,12 the q-Dunkl transform, 10 and the q-Bessel Fourier transform. [7][8][9] 3.1 q-Fourier-cosine transform  q Definition 2 (Bouzeffour 11 and Koornwinder and Swarttouw 12 ).…”

“…Various mathematical aspects of the uncertainty principle are studied; theorems of Hardy, Heisenberg‐Pauli‐Weyl type uncertainty inequalities and so forth are established for these transformations; see previous works 8,15,17,20,24,25,34,35 …”

“…Generalizations of this result in both classical and quantum analysis have been treated and many versions of Heisenberg-Pauli-Weyl type uncertainty inequalities have been obtained for several generalized Fourier transforms, see previous studies. [15][16][17][18][19][20][21][22][23][24][25][26][27][28] Recently, S. Ghobber 18 proved a variation on Heisenberg's uncertainty inequality for a wide variety of integral operators, including the Fourier transform, the Fourier-Bessel transform, the generalized Fourier transform and the G-transform. The proof of the new Heisenberg's uncertainty inequality follows by combining the Nash and Carlson's inequalities for these transformations.…”

“…Every discussion of the uncertainty principle must necessarily begin with the classical uncertainty principle which also called the Heisenberg‐Pauli‐Weyl uncertainty inequality 14 in which concentration is measured in terms of dispersions. This leads to this quantitative formulation in the form of a lower bound of the product of the dispersions of a nonzero signal f and its Fourier transform : where for Generalizations of this result in both classical and quantum analysis have been treated and many versions of Heisenberg‐Pauli‐Weyl type uncertainty inequalities have been obtained for several generalized Fourier transforms, see previous studies 15–28 …”

The composition of linear canonical wavelet transforms on generalized function spaces

“…Moreover, Srivastava et al [38] studied the fractional wavelet transformation and discussed some of its basic properties. Recently, continuous wavelet transform is one of the great achievements in the field of harmonic analysis [23,42]. The continuous wavelet transform can effectively useful in the study of signal analysis, image processing, pattern recognition and it also can solve many difficult problems that cannot be solved by using Fourier transform and Laplace transform.…”