In the present paper, we present a generalization of the \(\mathcal {L}\) -tensor product ( \(*_\mathcal {L}\) -product) which was introduced by Aeron et all. [2]. This includes a generalization of the well-known tensor cosine product and T-product defined for third-order tensors and based on discrete cosine transform (DCT) and fast Fourier transform, respectively. We give some theoretical results. As applications, we show how to use the new product for tensor completion on color images or color videos. In that case, the Proximal Gradient Algorithm (PGA) is used to solve some derived optimization problems. Numerical tests are given to show the effectiveness of the proposed methods as compared to well-known ones in the literature. To speed up the execution time of the new methods, we use GPU computation.