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

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Lebesgue Constants

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1.8.8 $$L_n=\frac{1}{\pi }{\int }_0^{\pi }\frac{\left|\sin \left(n+\frac{1}{2}\right)t\right|}{\sin \left(\frac{1}{2}t\right)}dt,$$ $n=0,1,\mathrm{\dots }$.

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Defines:

$L_n$: Lebesgue constants (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.8.E8

Encodings:

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See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $L_n$: Lebesgue constants

Permalink:

http://dlmf.nist.gov/1.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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Riemann–Lebesgue Lemma

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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 $ℝ$

… ►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

… ►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

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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 $x\in ℝ$. …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

… ►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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External Links:

ISSN 0030-8730, MathReview (Paolo M. Soardi), ZentralBlatt

Referenced by:

§1.8(i).

Encodings:

BibTeX

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

10: 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 $n$. …