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We first extend the famous sketch-and-project method and its adaptive variants for matrix linear systems to tensor linear systems with respect to the popular T-product of tensors. Their Fourier domain versions are also investigated. Then, considering that the existing sketching tensor or the way for sampling has some limitations, we propose two improved strategies. Convergence analysis for all the methods mentioned above are provided. We compare our methods with the existing ones usingdoi:10.48550/arxiv.2201.00667 fatcat:a3tof3r6e5cw3cknn6ar4xi66m