General solutions of the leakage in integral transforms and applications to the EB-leakage and detection of the cosmological gravitational wave background

For an orthogonal integral transform with complete dataset, any two components are linearly independent; however, when some data points are missing, there is going to be leakage from one component to another, which is referred to as the "leakage in integral transforms" in this work. A special case of this kind of leakage is the EB-leakage in detection of the cosmological gravitational wave background (CGWB). I first give the general solutions for all integral transforms, prove that they are the best solutions, and then apply them to the case of EB-leakage and detection of the CGWB. In the upcoming decade, most likely, new cosmic microwave background (CMB) data are from ground/balloon experiments, so they provide only partial sky coverage. Even in a fullsky mission, due to the Galactic foreground, part of the sky is still unusable. Within this context, the EB-leakage becomes inevitable. I show how to use the general solutions to achieve the minimal error bars of the EB-leakage, and use it to find out the maximum ability to detect the CGWB through CMB. The results show that, when focusing on the tensor-to-scalar ratio $r$ (at a pivot scale of 0.05 Mpc$^{-1}$), $1\%$ sky coverage ($f_{sky}=1\%$) is enough for a $5\sigma$-detection of $r\ge 10^{-2}$, but is barely enough for $r=10^{-3}$. If the target is to detect $r\sim 10^{-4}$ or $10^{-5}$, then $f_{sky}\ge 10\%$ is strongly recommended to enable a $5\sigma$-detection and to reserve some room for other errors.