# Stieltjes measure

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## 8 matching pages

##### 1: 1.1 Special Notation
 $x,y$ real variables. … the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$. …
##### 2: 1.4 Calculus of One Variable
###### StieltjesMeasure with $\alpha(x)$ Discontinuous
See Riesz and Sz.-Nagy (1990, Ch. 3). … Definite integrals over the Stieltjes measure $\,\mathrm{d}\alpha(x)$ could represent a sum, an integral, or a combination of the two. …
##### 3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
For a Lebesgue–Stieltjes measure $\,\mathrm{d}\alpha$ on $X$ let $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$,
1.18.13 $c_{n}=\left\langle f,\phi_{n}\right\rangle=\int_{a}^{b}f(x)\overline{\phi_{n}(% x)}\,\mathrm{d}x,$ $f\in L^{2}\left(X\right)$,
1.18.14 $\int_{a}^{b}{\left|f(x)\right|}^{2}\,\mathrm{d}x=\sum_{n=0}^{\infty}{\left|c_{% n}\right|}^{2},$ $f\in L^{2}\left(X\right)$,
1.18.64 $f(x)=\int_{\boldsymbol{\sigma}_{c}}\widehat{f}(\lambda)\phi_{\lambda}(x)\,% \mathrm{d}\lambda+\sum_{\boldsymbol{\sigma}_{p}}\widehat{f}(\lambda_{n})\phi_{% \lambda_{n}}(x),$ $f(x)\in C(X)\cap L^{2}\left(X\right)$.
##### 4: 18.2 General Orthogonal Polynomials
More generally than (18.2.1)–(18.2.3), $w(x)\,\mathrm{d}x$ may be replaced in (18.2.1) by $\,\mathrm{d}\mu(x)$, where the measure $\mu$ is the Lebesgue–Stieltjes measure $\mu_{\alpha}$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that $\int_{a}^{b}|x|^{n}\,\mathrm{d}\mu(x)<\infty$ for all $n$. …
##### 5: 1.16 Distributions
More generally, for $\alpha\colon[a,b]\to[-\infty,\infty]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure $\mu_{\alpha}$ (see §1.4(v)) can be considered as a distribution: … Since $\delta_{x_{0}}$ is the Lebesgue–Stieltjes measure $\mu_{\alpha}$ corresponding to $\alpha(x)=H\left(x-x_{0}\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$. … 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). …
##### 6: 18.39 Applications in the Physical Sciences
The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to-1{-}$. …
##### 7: Errata
The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
• ##### 8: 18.27 $q$-Hahn Class
For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …