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  "body": "\n## Summary\n\nA [[kernel|9M0NNNFJGM]] $K(x, y)$ of an [[integral transform|YCK709QB06]] is symmetric when it does not change if $x$ and $y$ are swapped.\nIt is self-reciprocal (a Fourier kernel) when the transform is its own inverse: applying it twice gives back the original function.\nThe Fourier cosine and sine kernels and the [[Hankel transform|2P9YWDGKC4]] kernel $\\sqrt{xy}\\, J_\\nu(xy)$ are both symmetric and self-reciprocal.\nFor such kernels the transform and its [[inverse transform|GC96MQC0PP]] are given by the same formula, which makes the theory and the tables symmetric.\n\n## Explanations\n\n### L1 human\n\nA symmetric kernel is like a symmetric matrix.\nA self-reciprocal kernel is like a matrix whose square is the identity, such as a reflection.\nFor the cosine kernel $\\sqrt{2/\\pi}\\cos(xy)$ on $(0, \\infty)$, if $g$ is the transform of $f$, then $f$ is the transform of $g$.\nSo a table of cosine transforms can be read from left to right or from right to left.\n",
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