In this paper, we consider the operator \(\Xi _{\alpha }^{\mathcal {M}}\) defined by \(\begin{aligned} \Xi _{\alpha }^{\mathcal {M}}f(x)=f'(-x)+\frac{\alpha }{x}(f(x)-f(-x))+\frac{ia}{b}xf(-x),\,\alpha \geqslant 0,\, a,b\in \mathbb {R},\,b\ne 0, \end{aligned}\) where f is a differentiable function on \(\mathbb {R}\) . We develop an in-depth harmonic analysis associated with the operator \(\Xi _{\alpha }^{\mathcal {M}}\) . Firstly, we introduce and analyze the linear canonical Hartley–Bessel transform, deriving some of its fundamental properties, such as inversion formula and Plancherel formula. Secondly, we present the translation operator associated with \(\Xi _{\alpha }^{\mathcal {M}}\) and explore some of its key properties. Next, we derive a convolution product for this transform. Building on the previous results, we define and study the linear canonical Hartley–Bessel wavelet transform \(\mathcal {W}^{\mathcal {M}}_{\psi }\) and establish its fundamental properties. Also some inequalities of this transform are proved. Finally, we define the localization operators \(\mathfrak {L}_{u,v}(\sigma )\) associated with \(\mathcal {W}^{\mathcal {M}}_{\psi }\) and we study the boundedness and compactness of these operators and establish a trace formula.
