square-integrable function

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1: 1.4 Calculus of One Variable

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Square-Integrable Functions

►A function $f(x)$ is square-integrable if …

2: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

3: 30.4 Functions of the First Kind

… ►If $f(x)$ is mean-square integrable on $[-1,1]$, then formally …

4: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

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§33.22(iv) Klein–Gordon and Dirac Equations

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Eigenstates using complex-rotated coordinates $r\to r{\mathrm{e}}^{\mathrm{i}\theta }$, so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

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5: 1.8 Fourier Series

… ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►

§1.8(iii) Integration and Differentiation

… ►when $f(x)$ and $g(x)$ are square-integrable and $a_n,b_n$ and $a_n^{\prime },b_n^{\prime }$ are their respective Fourier coefficients. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $|f^{\prime \prime }(x)|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. … ►Suppose also that $f(x)$ is integrable on $[0,\mathrm{\infty })$ and $f(x)\to 0$ as $x\to \mathrm{\infty }$. …