This paper investigates the Schur product of constacyclic codes via the constacyclic discrete Fourier transform (DFT). We first characterize key properties of the constacyclic DFT, highlighting its differences from the ordinary DFT. We then extend the concept of degenerate cyclic codes to constacyclic codes possessing a nontrivial pattern polynomial, thereby facilitating the analysis of their dimension sequences. Building on these tools, we generalize two established methods for computing the square of cyclic codes to compute the Schur product of arbitrary constacyclic codes. Finally, exploiting the inherent combinatorial structure, we derive properties of the Schur product dimension directly from additive combinatorics.