# pointwise

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## 4 matching pages

##### 1: 18.40 Methods of Computation

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►The question is then: how is this possible given only ${F}_{N}(z)$, rather than $F(z)$ itself? ${F}_{N}(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im}z$ at a distance greater than the pole spacing of the ${x}_{n}$, this may then be followed by

*approximate*numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to ${F}_{N}(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …##### 2: 1.9 Calculus of a Complex Variable

##### 3: 18.2 General Orthogonal Polynomials

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►This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with ${p}_{n}(x)={T}_{n}\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1.
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##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f({x}_{0}-)+f({x}_{0}+))/2$ if ${x}_{0}$ is an isolated point of discontinuity.
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