Two instances come to mind in digital signal processing (applied mathematics). The Fast Fourier Transform (FFT) computes the Digital Fourier Transform in $O(N \log N)$ instead of $O(N^2).$ Supposedly Gauss had a version of the FFT, long before (electronic) computers made their impact. The second is the original wavelet transform, by A. Haar in 1909. Research in wavelet transforms has exploded since.
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