# Lebesgue constants

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## 6 matching pages

##### 1: 1.8 Fourier Series
###### LebesgueConstants
1.8.8 $L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\mathrm{d}t,$ $n=0,1,\dots$.
1.8.9 $L_{n}\sim(4/\pi^{2})\ln n;$
##### 2: 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. …
##### 3: Bibliography F
• C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
• ##### 4: 2.4 Contour Integrals
Then … … Then the Laplace transform …where $\sigma$ ($\geq c$) is a constant. … If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) …
##### 5: 2.10 Sums and Sequences
Formula (2.10.2) is useful for evaluating the constant term in expansions obtained from (2.10.1). …where $\gamma$ is Euler’s constant5.2(ii)) and $\zeta'$ is the derivative of the Riemann zeta function (§25.2(i)). $e^{C}$ is sometimes called Glaisher’s constant. … where $\alpha$ ($\neq-1$) is a real constant, and … Hence by the Riemann–Lebesgue lemma (§1.8(i)) …
##### 6: 2.5 Mellin Transform Methods
(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) … where $l$ ($\geq 2$) is an arbitrary integer and $\delta$ is an arbitrary small positive constant. … … where $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right)$. …where $\gamma$ is Euler’s constant5.2(ii)). …