This is my first question here so pardon me if there are any mistakes.
In DSP course in Engineering, our professor stated the following
Even when neither condition for existence of the CTFT is satisfied, we may still be able to define a Fourier transform through a limiting process.
Let $x_n(t), n = 0,1,2,...$ denote a sequence of functions each of which has a valid CTFT $X_n(f)$.
Suppose that $\lim_{n\to\infty} x_n(t) = x(t),$ a function that does not have a valid transform.
If $X(f) = \lim_{n\to\infty} X_n(f)$ exists, we call it the generalized Fourier transform of $x(t)$
I'm very skeptical about this definition, all my research so far didn't land any reference that states this definition (extension) directly, except for Dirac delta (not) function, which makes sense since Dirac delta itself is defined through a limit (not necessarily) and the structure of both Dirac delta and Fourier transform is preserved under this definition.
Here are some of my questions about the definition
- Is it unique?
Suppose that $x(t) = \lim_{n\to\infty} = \lim_{n\to\infty}y_n,$ where $x_n \ne y_n$, does it still follow that $X(f) = \lim_{n\to\infty} X_n(f) = \lim_{n\to\infty} Y_n(f)$ for any $x_n, y_n$?
- Is it a valid extension?
Suppose that $x(t)$ has a valid Fourier transform, does it follow that for any $x_n(t)$ such that $x(t) = \lim_{n\to\infty} x_n(t)$, $X(f) = \lim_{n\to\infty} X_n(f)$?
Are the conditions stated above sufficient? That is for any function $x(t) = \lim_{n\to\infty} x_n(t)$ and $\lim_{n\to\infty} X_n(f)$ exists, $X(f)$ would be a valid transform?
I would be glad if you could name further resources (or a proof of the definition) to help me understand it.