Is there an accelerated algorithm or a discrete (Fourier) transform based method to convolve discrete-time signals sampled on a logarithmic grid? What I mean is representing a discrete-time signal by the sequence:

$$[1, 1, 1],$$

as a shorthand notation for samples:

$$\big[\big(\log(1), 1\big), \big(\log(2), 1\big), \big(\log(3), 1\big)\big],$$

with autoconvolution:

$$[1, 1, 1]*[1, 1, 1]$$

$$=\big[\big(\log(1)+\log(1), 1\big), \big(\log(1)+\log(2), 1\big), \big(\log(1)+\log(3), 1\big)\big]+\\ \big[\big(\log(2)+\log(1), 1\big), \big(\log(2)+\log(2), 1\big), \big(\log(2)+\log(3), 1\big)\big]+\\ \big[\big(\log(3)+\log(1), 1\big), \big(\log(3)+\log(2), 1\big), \big(\log(3)+\log(3), 1\big)\big]$$

$$\begin{align}=&\big[\big(\log(1\times1), 1\big), \big(\log(1\times2), 1\big), \big(\log(1\times3), 1\big)\big]\\ +&\big[\big(\log(2\times1), 1\big), \big(\log(2\times2), 1\big), \big(\log(2\times3), 1\big)\big]\\ +&\big[\big(\log(3\times1), 1\big), \big(\log(3\times2), 1\big), \big(\log(3\times3), 1\big)\big]\end{align}$$

$$\begin{align}=&\big[\big(\log(1), 1\big), \big(\log(2), 1\big), \big(\log(3), 1\big)\big]\\ +&\big[\big(\log(2), 1\big), \big(\log(4), 1\big), \big(\log(6), 1\big)\big]\\ +&\big[\big(\log(3), 1\big), \big(\log(6), 1\big), \big(\log(9), 1\big)\big]\end{align}$$

$$\begin{align}=&[1, 1, 1, 0, 0, 0, 0, 0, 0]\\ +&[0, 1, 0, 1, 0, 1, 0, 0, 0]\\ +&[0, 0, 1, 0, 0, 1, 0, 0, 1]\\ \rule{0pt}{4ex}=&[1, 2, 2, 1, 0, 2, 0, 0, 1]\end{align}$$

The sequence notation is not affected by the choice of base of the logarithm. In an alternative representation, a sample $(\log(0),1) $ $=$ $(-\infty,1)$ could be added to the signal (appended to the beginning of the sequence), giving the pleasures of zero-based indexing without affecting the rest of the convolution result:

$$[1,1,1,1]*[1,1,1,1]$$ $$\begin{align}=&[4, 0, 0, 0, 0, 0, 0, 0, 0, 0]\\ +&[1, 1, 1, 1, 0, 0, 0, 0, 0, 0]\\ +&[1, 0, 1, 0, 1, 0, 1, 0, 0, 0]\\ +&[1, 0, 0, 1, 0, 0, 1, 0, 0, 1]\\ \rule{0pt}{4ex}=&[7, 1, 2, 2, 1, 0, 2, 0, 0, 1]\end{align}$$

I don't have a real application mind, but encountered such a convolution in the context of multiplication of discrete random variables. It is also interesting that if an infinite sequence of 1's would be log-sampling-convolved by itself, then the result would be the sequence OEIS A000005, the number of divisors of positive integers.

Is there a practical accelerated algorithm or a theoretical discrete (Fourier) transform based method to convolve discrete-time signals sampled on a logarithmic grid? What I mean is representing a discrete-time signal by the sequence:

$$[1, 1, 1],$$

as a shorthand notation for samples:

$$\big[\big(\log(1), 1\big), \big(\log(2), 1\big), \big(\log(3), 1\big)\big],$$

with autoconvolution:

$$[1, 1, 1]*[1, 1, 1]$$

$$=\big[\big(\log(1)+\log(1), 1\big), \big(\log(1)+\log(2), 1\big), \big(\log(1)+\log(3), 1\big)\big]+\\ \big[\big(\log(2)+\log(1), 1\big), \big(\log(2)+\log(2), 1\big), \big(\log(2)+\log(3), 1\big)\big]+\\ \big[\big(\log(3)+\log(1), 1\big), \big(\log(3)+\log(2), 1\big), \big(\log(3)+\log(3), 1\big)\big]$$

$$\begin{align}=&\big[\big(\log(1\times1), 1\big), \big(\log(1\times2), 1\big), \big(\log(1\times3), 1\big)\big]\\ +&\big[\big(\log(2\times1), 1\big), \big(\log(2\times2), 1\big), \big(\log(2\times3), 1\big)\big]\\ +&\big[\big(\log(3\times1), 1\big), \big(\log(3\times2), 1\big), \big(\log(3\times3), 1\big)\big]\end{align}$$

$$\begin{align}=&\big[\big(\log(1), 1\big), \big(\log(2), 1\big), \big(\log(3), 1\big)\big]\\ +&\big[\big(\log(2), 1\big), \big(\log(4), 1\big), \big(\log(6), 1\big)\big]\\ +&\big[\big(\log(3), 1\big), \big(\log(6), 1\big), \big(\log(9), 1\big)\big]\end{align}$$

$$\begin{align}=&[1, 1, 1, 0, 0, 0, 0, 0, 0]\\ +&[0, 1, 0, 1, 0, 1, 0, 0, 0]\\ +&[0, 0, 1, 0, 0, 1, 0, 0, 1]\\ \rule{0pt}{4ex}=&[1, 2, 2, 1, 0, 2, 0, 0, 1]\end{align}$$

The sequence notation is not affected by the choice of base of the logarithm. In an alternative representation, a sample $(\log(0),1) $ $=$ $(-\infty,1)$ could be added to the signal (appended to the beginning of the sequence), giving the pleasures of zero-based indexing without affecting the rest of the convolution result:

$$[1,1,1,1]*[1,1,1,1]$$ $$\begin{align}=&[4, 0, 0, 0, 0, 0, 0, 0, 0, 0]\\ +&[1, 1, 1, 1, 0, 0, 0, 0, 0, 0]\\ +&[1, 0, 1, 0, 1, 0, 1, 0, 0, 0]\\ +&[1, 0, 0, 1, 0, 0, 1, 0, 0, 1]\\ \rule{0pt}{4ex}=&[7, 1, 2, 2, 1, 0, 2, 0, 0, 1]\end{align}$$

I don't have a real application mind, but encountered such a convolution in the context of multiplication of discrete random variables. It is also interesting that if an infinite sequence of 1's would be log-sampling-convolved by itself, then the result would be the sequence OEIS A000005, the number of divisors of positive integers.