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If I have the discrete Hartley transform of a signal of length 2n, how can I use it to efficiently obtain the discrete Hartley transform of the same signal padded to length 2n+m for some m > 0?

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Since the convolution theorem applies to the Hartley transform as well as the DFT, you may be able to use the same technique as can be used for FFT interpolation or spectrum resampling. Use the Hartley transform of the implied rectangular window as an interpolation kernel, and interpolate to a larger N. Perhaps window the interpolation kernel for a trade-off between accuracy and computational efficiency.

Depending on the size and efficiency of the interpolation needed to meet your requirements, it may be more efficient to do an inverse and forward transform instead.

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