Convolution, Fourier transform and Sobolev spaces generated by non-local Ionkin problem

In this work, given a second order differential operator $\mathcal B$ subject to non-local boundary conditions, we assign Fourier transform and convolution to this problem. We study the properties of the introduced convolution and describe the class of test functions. We also introduce Sobolev spaces and obtain Plancherel identity related to operator $\mathcal B$.