The Fourier transform and its inverse are definite integrals:
$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi i x \xi} dx$$ $$f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2 \pi i \xi x} d\xi$$
However I can find no reference to the indefinite counterpart of these integrals, i.e.:
$$\int f(x) e^{-2 \pi i x \xi} dx \text{ }\text{ and } \int \hat{f}(\xi) e^{2 \pi i \xi x} d\xi$$
What is the meaning of each of these indefinite integrals? Can we make statements about $f(x)$ from either of these, and vice versa? Are there any good references discussing these antiderivative functions and their properties?
Do you know that $$g(x) = \int_{-A}^{A} \hat{f}(\xi) e^{2i \pi \xi x} d \xi = f \ast h(x)$$ where $h(x) = \frac{\sin(\pi x A)}{\pi x}$ is an ideal low-pass filter and $\ast$ is the convolution ?