Lebesgue%20constants

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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^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,

n=0,1,\dots

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Defines:: $L_{n}$ : Lebesgue constants (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.8.E8
Encodings:: pMML, png, TeX
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, $\ln\NVar{z}$ : 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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2: 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

\,\mathrm{d}\alpha

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. …

3: 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. …

4: 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

\,\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 …

5: Bibliography F

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FDLIBM (free C library)

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Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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C. L. Frenzen and R. Wong (1986) 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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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

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6: 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. …

7: 1.1 Special Notation

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

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8: 8.26 Tables

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•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

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Zhang and Jin (1996, Table 3.8) tabulates $\gamma\left(a,x\right)$ for $a=0.5,1,3,5,10,25,50,100$ , $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

9: 20 Theta Functions

Chapter 20 Theta Functions

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10: 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. …