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: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Let $X=[a,b]$ or $[a,b)$ or $(a,b]$ or $(a,b)$ be a (possibly infinite, or semi-infinite) interval in $ℝ$ . For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$ , … ►

1.18.13 $$c_n=⟨f,{\varphi }_n⟩={\int }_a^{b}f(x)\overline{{\varphi }_n(x)}dx,$$ $f\in L^2\left(X\right)$ ,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E13

Encodings:

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See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.14 $${\int }_a^b{\left|f(x)\right|}^{2}dx=\sum _{n=0}^{\mathrm{\infty }}{\left|c_n\right|}^2,$$ $f\in L^2\left(X\right)$ ,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E14

Encodings:

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See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.52 $$⟨{\left(z-T\right)}^{-1}f,f⟩={\int }_{𝝈}{\left|\widehat{f}(\lambda )\right|}^2\frac{d\lambda }{z-\lambda },$$ $f\in L^2\left(X\right)$ , $z\notin {𝝈}_c$ ,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral, $\notin$: not an element of, $z$: variable and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.18(vi), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E52

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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1.18.65 $$f(x)={\int }_{{𝝈}_c}\widehat{f}(\lambda ){\varphi }_{\lambda }(x)d\lambda +\sum _{{𝝈}_p}\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x),$$ $f(x)\in C(X)\cap L^2\left(X\right)$ .

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $\cap$: intersection, $C(I)$ or $C(a,b)$: continuous on an interval $I$ or $(a,b)$ and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.18.E65

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(viii), §1.18 and Ch.1

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3: 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 …

4: 30.4 Functions of the First Kind

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

5: 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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6: 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 $x\to \mathrm{\infty }$. …