The trouble with Fourier transforms is that people use different definitions. The one I prefer is . But some people put a factor of 2π (or its square root) outside the integral or in the exponential. From the answers that you have given, it looks as though your lecturer's definition is (with j instead of i as the square root of –1, since he is an engineer rather than a mathematician, and without the negative sign in the exponential). Is that correct? If so, then your answer to a) is correct. I think that the answer to b) should have an extra factor of 1/2 in the exponential: . I think that the answer to c) is correct (use a plain asterisk to denote a convolution, not a tensor product symbol: ).
Finally, I don't understand the comment "haven't expanded the Fourier series". These are Fourier transforms, not Fourier series (totally different topics).