Yes that's true. But if you come at it looking at it as a sampled continuous signal and spectrum, you don't think of the FFT as fast polynomial multiplication. Neither viewpoint is wrong. Just different lenses for looking at different problems.

I think I still think of it that way. In EE you are convolving a filter with a signal in the time domain, which you can think of as a polynomial. Doing the fft, then a vector multiply +ifft accomplishes the same thing, but using the fft.

How can you think of a continuous time signal as a polynomial though? I don't know of a continuous time analogy. It seems like the polynomial only works for the discrete case. And FIR filter convolution is rarely computed by FFT in FPGAs (in stream applications).

Look at a system doing DSP for example. Let's say you have an input RF signal that goes into an analog band-pass filter, a mixer, an analog low-pass filter, a DAC, and a FPGA. The FPGA implements a FIR filter - lets say a Hilbert transform using a distributed arithmetic algorithm. You are getting envelope information out of the FPGA. There's no FFT anywhere in the FPGA to compute this filter output.

You are seeing some weird noise in your DSP output. You dig a little by looking at the output of the DAC and then look at the FFT magnitude log10. You see some suspicious spurs popping up, but at really strange discrete frequencies. You show the spectrum to an RF engineer, then you two go to the lab and put an analog spectrum analyzer on the input to the DAC. Sure enough, there are mixing products of your input carrier, mixer frequency, and sampling clock frequency. The weird spur frequencies make sense now - you are seeing folding of the mixing products from the sampling. You tweak the sampling clock to move the spur digital frequencies.

In that long but real world example, I don't see anywhere here where thinking of things in terms of polynomials would be helpful here. It's also going to lead to confusion of terminology with your RF engineer - they tend to think continuous.

Sorry, I think I misspoke. I didn't mean that I think of it in those terms normally. working with RF everyday, I think of it as time and frequency domain in the sense that you were describing. But mathematically I could see how the polynomial makes sense. Also, I was only thinking about discreet time since we are talking about the fft.