I'm working on some operations relating to Fourier transforms. I would like to neatly combine the multiplication of these two integrals, preferrably grouping Y(w)'s or y(t)'s together somehow, y(t) being a real valued function and Y(w) being it's complex Fourier transform. Is there a way to do that without ending up with a convolution operator in the expression?
$$\int_{-\infty}^{\infty} Y(w) e^{\frac{-(w+b)^2}{4a}}e^{iwt} \;\text{d}w \int_{-\infty}^{\infty} Y(w) e^{\frac{-(w-b)^2}{4a}}e^{iwt} \;\text{d}w \;\;\; \text{ for } a,b\in \mathbb{R}\,.$$