Integral Transforms and Generalized Quotient Space on the Torus

In this chapter, we discuss one of the recent generalization of Schwartz distributions that has significantly influenced the expansion of various mathematical disciplines. Here, we study the space of generalized quotient on the torus. Different integral transforms are investigated on the space of generalized quotients on the torus \(\mathcal {B}_{\mathcal {S}^{\prime }}(T^{d})\) . The space \(\mathcal {B}_{\mathcal {S}^{\prime }}(T^{d})\) is made of both distributions as well as space of hyperfunctions on the torus. Further, by introducing the relation between the Fourier and other integral transforms, the conditional theorems are proved for generalized quotients on tours. Moreover, we study the convergence structure of delta-convergence on the generalized quotient space, and an inversion theorem is proved.