# Absolutely continuous integration measure

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## 4 matching pages

##### 1: 1.4 Calculus of One Variable

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###### Absolutely Continuous Stieltjes Measure

►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha $ is*absolutely continuous*if $\alpha (x)$ is continuous and there exists a*weight function*$w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that …##### 2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►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 $,
…When $\alpha $ is absolutely continuous, i.
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►for $f(x)\in {L}^{2}$ and piece-wise continuous, with convergence as discussed in §1.18(ii).
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►Eigenfunctions corresponding to the continuous spectrum are non-${L}^{2}$ functions.
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►For fixed angular momentum $\mathrm{\ell}$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues ${\lambda}_{n},n=0,1,\mathrm{\dots},N-1$, with corresponding ${L}^{2}([0,\mathrm{\infty}),{r}^{2}dr)$ eigenfunctions ${\varphi}_{n}(r)$, and also a continuous spectrum $\lambda \in [0,\mathrm{\infty})$, with Dirac-delta normalized eigenfunctions ${\varphi}_{\lambda}(r)$, also with measure
${r}^{2}dr$.
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##### 3: 1.16 Distributions

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►, a function $f$ on $I$ which is absolutely Lebesgue integrable on every compact subset of $I$) such that
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►If the measure
${\mu}_{\alpha}$ is absolutely continuous with density $w$ (see §1.4(v)) then $\mathit{D}\alpha ={\mathrm{\Lambda}}_{w}$.
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►A

*tempered distribution*is a continuous linear functional $\mathrm{\Lambda}$ on $\mathcal{T}$. … ►Tempered distributions are continuous linear functionals on this space of test functions. … ►See Hildebrandt (1938) and Chihara (1978, Chapter II) for Stieltjes measures which are used in §18.39(iii); see also Shohat and Tamarkin (1970, Chapter II). …##### 4: 18.39 Applications in the Physical Sciences

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►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty}).$
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►Orthogonality, with measure
$dr$ for $r\in [0,\mathrm{\infty})$, for fixed $l$
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►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure
$dr$.
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►is tridiagonalized in the complete ${L}^{2}$ non-orthogonal (with measure
$dr$, $r\in [0,\mathrm{\infty})$) basis of Laguerre functions:
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►Given that $a=-b$ in both the attractive and repulsive cases, the expression for the absolutely continuous, $x\in [-1,1]$, part of the function of (18.35.6) may be simplified:
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