Lebesgue constants

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11: 1.4 Calculus of One Variable

… ► … ► $c$ and $d$ constants. … ►

Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

… ►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. … …

12: 18.18 Sums

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18.18.1 $$a_n=\frac{n!(2n+\alpha +\beta +1)\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{2^{\alpha +\beta +1}\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{\int }_{-1}^{1}f(x)P_n^{(\alpha ,\beta )}\left(x\right){(1-x)}^{\alpha }{(1+x)}^{\beta }dx.$$

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Defines:

$a_n$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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18.18.5 $$b_n=\frac{n!}{\mathrm{\Gamma }\left(n+\alpha +1\right)}{\int }_0^{\mathrm{\infty }}f(x)L_n^{(\alpha )}\left(x\right){\mathrm{e}}^{-x}{x}^{\alpha }dx.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

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Expansion of $L^2$ functions

►In all three cases of Jacobi, Laguerre and Hermite, if $f(x)$ is $L^2$ on the corresponding interval with respect to the corresponding weight function and if $a_n,b_n,d_n$ are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in $L^2$ sense. … ►See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of $P_n^{(\gamma ,\delta )}\left(x\right)$ in terms of $P_n^{(\alpha ,\beta )}\left(x\right)$. …

13: 2.5 Mellin Transform Methods

… ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) … ►where $l$ ($\ge 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. … … ►where $\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right)$. …where $\gamma$ is Euler’s constant (§5.2(ii)). …