# Lebesgue

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##### 1: David M. Bressoud
Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …
##### 2: 1.8 Fourier Series
###### Lebesgue Constants
1.8.9 $L_{n}\sim(4/{\pi}^{2})\ln n;$
##### 3: 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 $\,\mathrm{d}\alpha$. …
##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(ii) $L^{2}$ spaces on intervals in $\mathbb{R}$
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 $\,\mathrm{d}\alpha$, …The space $L^{2}\left(X,\,\mathrm{d}\alpha\right)$ becomes a separable Hilbert space with inner product … Eigenfunctions corresponding to the continuous spectrum are non-$L^{2}$ functions. … The well must be deep and broad enough to allow existence of such $L^{2}$ discrete states. …
##### 5: 18.39 Applications in the Physical Sciences
Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^{2}$ and forming a complete set. … The spectrum is mixed, as in §1.18(viii), the positive energy, non-$L^{2}$, scattering states are the subject of Chapter 33. … 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. … 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. …
##### 6: 1.4 Calculus of One Variable
###### Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals
A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for $x\in\mathbb{R}$. …Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure $\,\mathrm{d}\mu(x)$ and it is well defined for functions $f$ which are integrable with respect to that more general measure. … … For $\alpha(x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $\,\mathrm{d}\alpha$ is absolutely continuous if $\alpha(x)$ is continuous and there exists a weight function $w(x)\geq 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that …
##### 7: 3.11 Approximation Techniques
to the maximum error of the minimax polynomial $p_{n}(x)$ is bounded by $1+L_{n}$, where $L_{n}$ is the $n$th Lebesgue constant for Fourier series; see §1.8(i). … Moreover, the set of minimax approximations $p_{0}(x),p_{1}(x),p_{2}(x),\dots,p_{n}(x)$ requires the calculation and storage of $\frac{1}{2}(n+1)(n+2)$ coefficients, whereas the corresponding set of Chebyshev-series approximations requires only $n+1$ coefficients. …
##### 8: Bibliography F
• C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
• ##### 9: 1.16 Distributions
, a function $f$ on $I$ which is absolutely Lebesgue integrable on every compact subset of $I$) such that …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 $\alpha$. …
##### 10: 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 $n$. …