# delta sequences

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## 1—10 of 17 matching pages

##### 1: 1.17 Integral and Series Representations of the Dirac Delta
###### §1.17(i) DeltaSequences
for a suitably chosen sequence of functions $\delta_{n}\left(x\right)$, $n=1,2,\dots$. Such a sequence is called a delta sequence and we write, symbolically,
1.17.4 $\lim_{n\to\infty}\delta_{n}\left(x\right)=\delta\left(x\right),$ $x\in\mathbb{R}$.
An example of a delta sequence is provided by …
##### 2: 3.9 Acceleration of Convergence
###### §3.9(iii) Aitken’s $\Delta^{2}$-Process
3.9.7 $t_{n}=s_{n}-\frac{(\Delta s_{n})^{2}}{\Delta^{2}s_{n}}=s_{n}-\frac{(s_{n+1}-s_% {n})^{2}}{s_{n+2}-2s_{n+1}+s_{n}}.$
3.9.9 $t_{n,2k}=\frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})},$ $n=0,1,2,\dots$,
##### 3: 30.15 Signal Analysis
30.15.7 $\int_{-\tau}^{\tau}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t=\Lambda_{n}\delta_{k,n},$
30.15.8 $\int_{-\infty}^{\infty}\phi_{k}(t)\phi_{n}(t)\,\mathrm{d}t=\delta_{k,n}.$
The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$. …
##### 4: 3.6 Linear Difference Equations
3.6.2 $a_{n}\Delta^{2}w_{n-1}+(2a_{n}-b_{n})\Delta w_{n-1}+(a_{n}-b_{n}+c_{n})w_{n-1}% =d_{n},$
##### 5: 3.8 Nonlinear Equations
Let $z_{1},z_{2},\dots$ be a sequence of approximations to a root, or fixed point, $\zeta$. …for all $n$ sufficiently large, where $A$ and $p$ are independent of $n$, then the sequence is said to have convergence of the $p$th order. … For real functions $f(x)$ the sequence of approximations to a real zero $\xi$ will always converge (and converge quadratically) if either: … Starting this iteration in the neighborhood of one of the four zeros $\pm 1,\pm\mathrm{i}$, sequences $\{z_{n}\}$ are generated that converge to these zeros. For an arbitrary starting point $z_{0}\in\mathbb{C}$, convergence cannot be predicted, and the boundary of the set of points $z_{0}$ that generate a sequence converging to a particular zero has a very complicated structure. …
##### 6: 17.12 Bailey Pairs
17.12.1 $\sum_{n=0}^{\infty}\alpha_{n}\gamma_{n}=\sum_{n=0}^{\infty}\beta_{n}\delta_{n},$
$\gamma_{n}=\sum_{j=n}^{\infty}\delta_{j}u_{j-n}v_{j+n}.$
A sequence of pairs of rational functions of several variables $(\alpha_{n},\beta_{n})$, $n=0,1,2,\dots$, is called a Bailey pair provided that for each $n\geqq 0$When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. …
##### 7: 4.4 Special Values and Limits
where $a$ ($\in\mathbb{C}$) and $\delta$ ($\in(0,\tfrac{1}{2}\pi]$) are constants. …
4.4.19 $\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln n\right)=% \gamma=0.57721\ 56649\ 01532\ 86060\dots,$
##### 8: 18.27 $q$-Hahn Class
Here $a,b$ are fixed positive real numbers, and $I_{+}$ and $I_{-}$ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. …In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). …
18.27.4 $\sum_{y=0}^{N}Q_{n}(q^{-y})Q_{m}(q^{-y})\genfrac{[}{]}{0.0pt}{}{N}{y}_{q}\frac% {\left(\alpha q;q\right)_{y}\left(\beta q;q\right)_{N-y}}{\left(\alpha q\right% )^{y}}=h_{n}\delta_{n,m},$ $n,m=0,1,\ldots,N$,
18.27.14 $\sum_{y=0}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{y}(aq)^{y}% }{\left(q;q\right)_{y}}=h_{n}\delta_{n,m},$ $0,
##### 9: 5.17 Barnes’ $G$-Function (Double Gamma Function)
When $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)$, …
5.17.6 $A=e^{C}=1.28242\;71291\;00622\;63687\;\ldots,$
##### 10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_{n}\}$ in $V$ (i. … of the Dirac delta distribution. … , for each $v\in V$ there is a sequence $\{v_{n}\}$ in $\mathcal{D}(T)$ such that $\left\|{v_{n}-v}\right\|\to 0$ as $n\to\infty$. … Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta\left(x-y\right)=\frac{1}{\sqrt{xy}}\delta\left(x-y\right)$, … These latter results also correspond to use of the $\delta\left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). …