What is meant by "duality"? As far as I know, this term is used for scenarios where to and fro value/or procedures are the same.
For example convolution in the time domain is equal to multiplication in the frequency domain, and convolution in the frequency domain is equal to multiplication in the time domain
Is the above example a case of duality?
Please explain with an example in the case of the discrete Fourier transform.
Duality is a general idea of some abstract reversible map that matches functions/concepts from one domain to another domain. There is a duality between pseudo-vectors and antisymmetric matrices in 3D. There is a duality between planar graphs when we replace cells←→vertices. And many more.
In terms of Fourier transform, the duality is that we can represent a signal (function) either in a time domain or in a frequency domain. There are properties of how operations in the time domain are dual to other operations in the frequency domain (like multiplication-convolution example). This duality allows us to freely jump between time and frequency domains as we please to make some steps easier to calculate/conceptualize.
For example, it's hard to understand how a complicated linear differential operator acts on a complicated signal staying in a time domain. However, in a frequency domain it is just a multiplication of signal with the kernel, which we can easily imagine and talk about amplification/silencing of certain frequencies.