I am reading a paper about SVD of a operator. I know a little about SVD of matrix, and eigenvalue decomposition of operator. But I totally don't know what does SVD of a operator mean. In the following text, could anyone tell me the expressions of $u_k$ or $v_k$ and how to get them? Thanks in advance.
Let $ f(x) = P(\varphi)(x) = \int_{-1}^1 \log(x-t)\varphi(t) dt,\quad x \in [3,5] $ Using the language of classical potential theory, P is the operator mapping the charge disctribution on the interval $[-1,1]$ to the induced potential created on the interval $[3,5]$
One can construct the SVD of the operator P (defined above), representing it in the form $ f(x) = P(\varphi)(x) = \sum_{k=1}^\infty \lambda_k(v_k,\varphi)u_k(x) $ where $u_k:[3,5]\rightarrow \mathbf{R}, v_k:[-1,1]\rightarrow\mathbf{R} $ are the $k$th left and right singular vectors, respectively, and $\lambda_k$ is the $k$th singular value.