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In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} as kernels of the transform. This was first introduced by Lokenath Debnath in 1964.[1][2][3][4]
The Hermite transform of a function F ( x ) {\displaystyle F(x)} is H { F ( x ) } = f H ( n ) = ∫ − ∞ ∞ e − x 2 H n ( x ) F ( x ) d x {\displaystyle H\{F(x)\}=f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}
The inverse Hermite transform is given by H − 1 { f H ( n ) } = F ( x ) = ∑ n = 0 ∞ 1 π 2 n n ! f H ( n ) H n ( x ) {\displaystyle H^{-1}\{f_{H}(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}
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