# Fejér kernel

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## 1—10 of 29 matching pages

##### 1: 1.15 Summability Methods
###### FejérKernel
1.15.17 $K_{n}(\theta)\to 0,$
##### 2: Bibliography W
• J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
• C. S. Whitehead (1911) On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\mathrm{ker}$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.
• ##### 3: 10.62 Graphs Figure 10.62.2: ker ⁡ x , kei ⁡ x , ker ′ ⁡ x , kei ′ ⁡ x , 0 ≤ x ≤ 8 . Magnify Figure 10.62.4: e x / 2 ⁢ ker ⁡ x , e x / 2 ⁢ kei ⁡ x , e x / 2 ⁢ N ⁡ ( x ) , 0 ≤ x ≤ 8 . Magnify
##### 4: 31.10 Integral Equations and Representations
###### Kernel Functions
The kernel $\mathcal{K}$ must satisfy …
###### Kernel Functions
The kernel $\mathcal{K}$ must satisfy … leads to the kernel equation …
##### 5: 10.67 Asymptotic Expansions for Large Argument
###### §10.67(i) $\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x,\operatorname{ker}_{\nu}x,% \operatorname{kei}_{\nu}x$, and Derivatives
10.67.1 $\operatorname{ker}_{\nu}x\sim e^{-x/\sqrt{2}}\left(\frac{\pi}{2x}\right)^{% \frac{1}{2}}\*\sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\cos\left(\frac{x}{% \sqrt{2}}+\left(\frac{\nu}{2}+\frac{k}{4}+\frac{1}{8}\right)\pi\right),$
The contributions of the terms in $\operatorname{ker}_{\nu}x$, $\operatorname{kei}_{\nu}x$, $\operatorname{ker}_{\nu}'x$, and $\operatorname{kei}_{\nu}'x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). …
10.67.14 $\operatorname{ker}x\operatorname{kei}'x-\operatorname{ker}'x\operatorname{kei}% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}-\frac{1}{8}\frac{1}% {x}+\frac{9}{64\sqrt{2}}\frac{1}{x^{2}}-\frac{39}{512}\frac{1}{x^{3}}+\frac{75% }{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$
10.67.15 $\operatorname{ker}x\operatorname{ker}'x+\operatorname{kei}x\operatorname{kei}'% x\sim-\frac{\pi}{2x}e^{-x\sqrt{2}}\left(\frac{1}{\sqrt{2}}+\frac{3}{8}\frac{1}% {x}-\frac{15}{64\sqrt{2}}\frac{1}{x^{2}}+\frac{45}{512}\frac{1}{x^{3}}+\frac{3% 15}{8192\sqrt{2}}\frac{1}{x^{4}}+\dotsb\right),$
##### 6: 10.61 Definitions and Basic Properties
When $\nu=0$ suffices on $\operatorname{ber}$, $\operatorname{bei}$, $\operatorname{ker}$, and $\operatorname{kei}$ are usually suppressed. Most properties of $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, and $\operatorname{kei}_{\nu}x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. …
$\operatorname{ker}_{-\nu}x=\cos\left(\nu\pi\right)\operatorname{ker}_{\nu}x-% \sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x,$
$\operatorname{kei}_{-\nu}x=\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x+% \cos\left(\nu\pi\right)\operatorname{kei}_{\nu}x.$
$\operatorname{ker}_{-n}x=(-1)^{n}\operatorname{ker}_{n}x,~{}\operatorname{kei}% _{-n}x=(-1)^{n}\operatorname{kei}_{n}x.$
##### 7: 10.63 Recurrence Relations and Derivatives
###### §10.63(i) $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, $\operatorname{kei}_{\nu}x$
$\operatorname{ker}_{\nu}x,\operatorname{kei}_{\nu}x;$
$\operatorname{kei}_{\nu}x,-\operatorname{ker}_{\nu}x.$
$\sqrt{2}\operatorname{ker}'x=\operatorname{ker}_{1}x+\operatorname{kei}_{1}x,$
Equations (10.63.6) and (10.63.7) also hold when the symbols $\operatorname{ber}$ and $\operatorname{bei}$ in (10.63.5) are replaced throughout by $\operatorname{ker}$ and $\operatorname{kei}$, respectively. …
##### 8: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …
##### 9: 10.70 Zeros
$\mbox{zeros of \operatorname{ker}_{\nu}x}\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi$,
##### 10: 13.27 Mathematical Applications
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …