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The purpose of this contribution is to introduce a method for constructing temporally adaptive, sparse, accurate, efficient and reliable representations of large scale, discretely sampled transient space-time electromagnetic wave fields, through the use of sparse wavelet approximations. The temporally adaptive sparse wavelet approximations are achieved by performing a three step computational process during a time domain electromagnetic simulation. The computational process consists of a forward Fast Wavelet Transform (FWT) step, an adaptive wavelet coefficient truncation step, and an inverse FWT step. The sparse approximation of a space-time electromagnetic wave field results in retaining wavelet coefficients with the largest magnitudes, localized in regions of the solution domain with high electromagnetic energy density concentrations. Numerical results that demonstrate the applicability and versatility of the approach are provided. Rigorous error analyses are also provided to demonstrate the accuracy of the method.