pointwise

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4 matching pages

1: 18.40 Methods of Computation

… ►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

\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

… ►This sequence converges pointwise to a function

f(z)

if …

3: 18.2 General Orthogonal Polynomials

… ►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. …

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►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

x_{0}

is an isolated point of discontinuity. …