# Lebesgue constants

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##### 1: 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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##### 2: 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. …##### 3: 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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##### 4: 2.4 Contour Integrals

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►Then
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►Then the Laplace transform
…where $\sigma $ ($\ge c$) is a constant.
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►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i))
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##### 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: 1.18 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. …##### 7: 1.16 Distributions

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►where ${\alpha}_{1}$ and ${\alpha}_{2}$ are real or complex constants.
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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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►where $c$ is a constant.
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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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##### 8: 2.10 Sums and Sequences

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►Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1).
…where $\gamma $ is Euler’s constant (§5.2(ii)) and ${\zeta}^{\prime}$ is the derivative of the Riemann zeta function (§25.2(i)).
${\mathrm{e}}^{C}$ is sometimes called

*Glaisher’s constant*. … ►where $\alpha $ ($\ne -1$) is a real constant, and … ►Hence by the Riemann–Lebesgue lemma (§1.8(i)) …##### 9: 1.5 Calculus of Two or More Variables

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►that is, for every arbitrarily small positive constant
$\u03f5$ there exists $\delta $ ($>0$) such that
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►In particular, ${\varphi}_{1}(x)$ and ${\varphi}_{2}(x)$ can be constants.
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►Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times (c,d)$, then
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►A more general concept of integrability (both finite and infinite) for functions on domains in ${\mathbb{R}}^{n}$ is

*Lebesgue integrability*. …##### 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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###### §18.2(iii) Standardization and Related Constants

… ►###### Constants

… ►(i) the*traditional OP*standardizations of Table 18.3.1, where each is defined in terms of the above constants. … ►, of the form $w(x)dx$) nor is it necessarily unique, up to a positive constant factor. …