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The sequence in L2 (R) of Hermite polynomials can be orthonormalized, this constitutes a basis for the L2 space, then, in analogy with the n-dimensional case, any function f can be approximated by a linear combination of this sequence. This code graphs the approximation of various functions in terms of the 'Hermite base' at L2 (R).

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functions-in-l2-in-terms-of-the-hermite-basis's Introduction

Functions in $\large L^2(\mathbb{R})$ in terms of the Hermite basis

The sequence in $\large L^2(\mathbb{R})$ (defined for all reals) of Hermite polynomials can be orthonormalized, this constitutes a basis for the $\large L^2(\mathbb{R})$ space, then, in analogy with the n-dimensional case, any function f can be approximated by a linear combination of this sequence. This code graphs the approximation of various functions in terms of the 'Hermite basis' at $\large L^2(\mathbb{R})$ . If

$f(x) \in L^2(\mathbb{R})\longrightarrow f=\sum_{n=0}^{\infty}\sigma_n\psi_n\\$

Where $\psi_n$ is the Hermite function of order n, given by:

$\large \psi_{n}(x) = \frac{1}{(2^{n}n!\sqrt{\pi})^{1/2}}e^{-\frac{x^2}{2}}H_{n}\quad n=0,1,2..,$

The last one constitutes a total orthonormal sequence in $\large L^2(\mathbb{R})$ , and the corresponding Hermite polynomial is given by: $H_0=1, \quad H_n = \sum_{j=0}^{N}n!\frac{(-1)^{j}2^{n-2j}}{j!(n-2j)!}x^{n-2j}, \quad n=1,2,3,...$

Using the properties of orthonormality of the sequence $\large \{\psi_n\}_n$ we can obtain the coefficients of the combination by taking the inner product

$\sigma_n = \langle f, \psi_n \rangle = \int_{-\infty}^{\infty} f \frac{1}{(2^{n}n!\sqrt{\pi})^{1/2}}e^{-\frac{x^2}{2}}H_n dx$

With this information we can obtain graphical representations of the approximations.

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krystifer / functions-in-l2-in-terms-of-the-hermite-basis Goto Github PK

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Functions in $\large L^2(\mathbb{R})$ in terms of the Hermite basis