Lebesgue–Stieltjes measure

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1: 1.1 Special Notation

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$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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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 $d\alpha$, … ►

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

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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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1.18.64 $$f(x)={\int }_{{𝝈}_c}\widehat{f}(\lambda ){\varphi }_{\lambda }(x)d\lambda +\sum _{{𝝈}_p}\widehat{f}({\lambda }_n){\varphi }_{{\lambda }_n}(x),$$ $f(x)\in C(X)\cap L^2\left(X\right)$.

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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.E64

Encodings:

pMML, png, TeX

See also:

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

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3: 18.2 General Orthogonal Polynomials

… ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, 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 for all $n$. …

4: 1.16 Distributions

… ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ 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 $\alpha$. …

5: 1.4 Calculus of One Variable

… ►Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure $d\mu (x)$ and it is well defined for functions $f$ which are integrable with respect to that more general measure. …

6: 18.39 Applications in the Physical Sciences

… ►with an infinite set of orthonormal $L^2$ eigenfunctions … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. …is tridiagonalized in the complete $L^2$ non-orthogonal (with measure $dr$, $r\in [0,\mathrm{\infty })$) basis of Laguerre functions: … ►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 $x\to -1-$. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …