Absolutely continuous integration measure

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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

… ►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. … ►for $f(x)\in L^2$ and piece-wise continuous, with convergence as discussed in §1.18(ii). … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►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 $r^{2}dr$. …

3: 1.16 Distributions

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

… ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. … ►is tridiagonalized in the complete $L^2$ non-orthogonal (with measure $dr$, $r\in [0,\mathrm{\infty })$) basis of Laguerre functions: … ►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: …