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  "body": "\n## Definition\n\nLet $a \u003c b$.\nA *Hermitian kernel* on $[a, b]$ is a [[kernel|GSMBXT7FPK]] $K \\colon [a, b] \\times [a, b] \\to \\C$ such that\n$$K(x, s) = \\overline{K(s, x)} \\quad \\text{for all } x, s \\in [a, b],$$\nwhere the bar is the [[complex conjugate|0FTH2E9TAS]].\n\n## Remarks\n\nExample: $K(x, s) = e^{i (x - s)}$.\nThe check: $\\overline{K(s, x)} = \\overline{e^{i (s - x)}} = e^{-i (s - x)} = e^{i (x - s)} = K(x, s)$.\n\nNon-example: $K(x, s) = i$, the constant kernel.\nHere $\\overline{K(s, x)} = -i \\ne i$.\n\nOn the diagonal a Hermitian kernel is real, since $K(x, x) = \\overline{K(x, x)}$.\nA real-valued kernel is Hermitian exactly when it is symmetric.\nFor square-integrable kernels the condition is often required only for almost all $(x, s)$.\nThen the integral operator $K$ on $L^2(a, b)$ satisfies $\\ip{K u}{v} = \\ip{u}{K v}$ for the inner product $\\ip{u}{v} = \\int_a^b u \\bar v \\, dx$.\n",
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