Applying a Filter Kernel Defined by a Continuous Equation in the Frequency Domain
Given an Image $ I \in \mathbb{R}^{m \times n} $ I would solve it as following (Using the Convolution Theorem):


I would sample the function in Fourier Domain into a grid of $ m \times n $.
I would apply DFT transform on the image to get the $ m \times n $ representation in the Fourier Domain.
I would apply Element Wise multiplication between the 2 $ m \times n $ arrays.
I would apply Inverse DFT on the result to get the filtered image in the spatial domain.


Pay attention that this will result in Cyclic Convolution and not Linear Convolution (Which requires more padding).