Two instances come to mind in digital signal processing (applied mathematics).
The Fast Fourier Transform (FFT) computes the DigitalDiscrete 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.