From the reference by Knutsen, page 25, Kernel linear independence test is explained
Knutsen, Sverre. "Gaussian processes for online system identification and control of a quadrotor." (2019).
which is actually from the reference by Csató
Csató, Lehel, and Manfred Opper. "Sparse on-line Gaussian processes." Neural Computation 14.3 (2002): 641-668.
The author presented the equation 2.41 which is $$ k({\mathbf{x}},{\mathbf{x}}_{t+1} ) \approx \sum_{i=1}^{N_{BV}} \alpha_i k({\mathbf{x}},{\mathbf{x}}_{i} ). $$
where $$ \boldsymbol{\alpha}=\mathbf{K}^{-1}\mathbf{y} $$ where $\mathbf{K}$ is a covariance matrix from a kernel function with output $\mathbf{y}=[\mathbf{y}_1,...,\mathbf{y}_{N_{BV}}]$ and input $\mathbf{X}=[\mathbf{x}_1,...,\mathbf{x}_{N_{BV}}]$.
The equation states that a kernel function $k(\mathbf{x},\mathbf{x}_{t+1})$ at some new sample point $\mathbf{x}_{t+1}$ can be described by the old kernels in the function space. Would you explain how the author derived this approximation?