Attempting to model the diffraction for a black-body source, I have stumbled upon a bit of a conundrum with the mathematics. I am not asking WHY the light is diffracted, but instead looking for a detailed analysis of the diffraction itself.
Modelling the diffraction for an image is also well understood in the theory of convolution. Assuming we have an image from a "perfect" instrument, that perhaps has noise unrelated to diffraction, then we can convolve the image with a function representing the diffraction pattern of the instrument (the point spread function or PSF), see image below.
The process is mathematically described by a convolution equation of the form:
$$ \text{image} = \text{object} \circledast \text{PSF} $$
What I do not quite understand is what exactly we are convolving when it comes to a black-body source. A black-body emits light for all wavelengths, and our PSF, say for example the Airy disc, is itself wavelength-dependent.
From my understanding of this then, to correctly model the diffraction, first we must convolve the radiance with the PSF (say from a circular aperture), and only then integrate the radiance for the bandwidth of wavelengths.
i.e. : $$B_{\lambda}(\lambda,T) \circledast \text{PSF} = \frac{2\pi hc^2}{\lambda^5}\dfrac{1}{\exp\left(\frac{hc}{k_BT\lambda}\right) -1 } \circledast \left [ \frac{2 J_1(k\,a \sin \theta)}{k\,a \sin \theta} \right ]^2 $$
and so over a range of wavelengths, I expect : $$ \int_{\lambda_1}^{\lambda_2}( B(\lambda,T) \circledast \text{PSF} ) d\lambda = \int_{\lambda_1}^{\lambda_2}\left(\frac{2\pi hc^2}{\lambda^5}\dfrac{1}{\exp\left(\frac{hc}{k_BT\lambda}\right) -1 } \circledast \left [ \frac{2 J_1(k\,a \sin \theta)}{k\,a \sin \theta} \right ]^2\right)d\lambda$$
... essentially, I am asking is the maths above sound and physically correct?
