# Lebesgue

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## 1—10 of 16 matching pages

##### 1: David M. Bressoud

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► 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

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###### Lebesgue Constants

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1.8.8
$${L}_{n}=\frac{1}{\pi}{\int}_{0}^{\pi}\frac{\left|\mathrm{sin}\left(n+\frac{1}{2}\right)t\right|}{\mathrm{sin}\left(\frac{1}{2}t\right)}dt,$$
$n=0,1,\mathrm{\dots}$.

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1.8.9
$${L}_{n}\sim (4/{\pi}^{2})\mathrm{ln}n;$$

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###### Riemann–Lebesgue Lemma

…##### 3: 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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##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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###### §1.18(ii) ${L}^{2}$ spaces on intervals in $\mathbb{R}$

… ►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 $, …The space ${L}^{2}(X,d\alpha )$ 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

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►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.
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►The spectrum is mixed, as in §1.18(viii), the positive energy, non-${L}^{2}$, scattering states are the subject of Chapter 33.
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►with an infinite set of orthonormal ${L}^{2}$ eigenfunctions
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►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}$) in Chapter 33, where the solutions appear as Whittaker functions.
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►The fact that non-${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

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###### 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*$d\mu (x)$ 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 $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 …##### 7: 3.11 Approximation Techniques

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►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 ${p}_{0}(x),{p}_{1}(x),{p}_{2}(x),\mathrm{\dots},{p}_{n}(x)$ 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

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Asymptotic expansions of the Lebesgue constants for Jacobi series.
Pacific J. Math. 122 (2), pp. 391–415.
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##### 9: 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
…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:
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►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 $.
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##### 10: 18.2 General Orthogonal Polynomials

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►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 $n$.
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