Stable Extrapolation of Analytic Functions
Abstract This paper examines the problem of extrapolation of an analytic function for $$x > 1$$ x...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131505 |
| _version_ | 1811085670691635200 |
|---|---|
| author | Demanet, Laurent Townsend, Alex |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Demanet, Laurent Townsend, Alex |
| author_sort | Demanet, Laurent |
| collection | MIT |
| description | Abstract
This paper examines the problem of extrapolation of an analytic function for
$$x > 1$$
x
>
1
given
$$N+1$$
N
+
1
perturbed samples from an equally spaced grid on
$$[-1,1]$$
[
-
1
,
1
]
. For a function f on
$$[-1,1]$$
[
-
1
,
1
]
that is analytic in a Bernstein ellipse with parameter
$$\rho > 1$$
ρ
>
1
, and for a uniform perturbation level
$$\varepsilon $$
ε
on the function samples, we construct an asymptotically best extrapolant e(x) as a least squares polynomial approximant of degree
$$M^*$$
M
∗
determined explicitly. We show that the extrapolant e(x) converges to f(x) pointwise in the interval
$$I_\rho \in [1,(\rho +\rho ^{-1})/2)$$
I
ρ
∈
[
1
,
(
ρ
+
ρ
-
1
)
/
2
)
as
$$\varepsilon \rightarrow 0$$
ε
→
0
, at a rate given by a x-dependent fractional power of
$$\varepsilon $$
ε
. More precisely, for each
$$x \in I_{\rho }$$
x
∈
I
ρ
we have
$$\begin{aligned} |f(x) - e(x)| = \mathcal {O}\left( \varepsilon ^{-\log r(x) / \log \rho } \right) , \quad r(x) = \frac{x+\sqrt{x^2-1}}{\rho }, \end{aligned}$$
|
f
(
x
)
-
e
(
x
)
|
=
O
ε
-
log
r
(
x
)
/
log
ρ
,
r
(
x
)
=
x
+
x
2
-
1
ρ
,
up to log factors, provided that an oversampling conditioning is satisfied, viz.
$$\begin{aligned} M^* \le \frac{1}{2} \sqrt{N}, \end{aligned}$$
M
∗
≤
1
2
N
,
which is known to be needed from approximation theory. In short, extrapolation enjoys a weak form of stability, up to a fraction of the characteristic smoothness length. The number of function samples does not bear on the size of the extrapolation error provided that it obeys the oversampling condition. We also show that one cannot construct an asymptotically more accurate extrapolant from equally spaced samples than e(x), using any other linear or nonlinear procedure. The proofs involve original statements on the stability of polynomial approximation in the Chebyshev basis from equally spaced samples and these are expected to be of independent interest. |
| first_indexed | 2024-09-23T13:13:25Z |
| format | Article |
| id | mit-1721.1/131505 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:13:25Z |
| publishDate | 2021 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1315052023-09-14T20:14:16Z Stable Extrapolation of Analytic Functions Demanet, Laurent Townsend, Alex Massachusetts Institute of Technology. Department of Mathematics Abstract This paper examines the problem of extrapolation of an analytic function for $$x > 1$$ x > 1 given $$N+1$$ N + 1 perturbed samples from an equally spaced grid on $$[-1,1]$$ [ - 1 , 1 ] . For a function f on $$[-1,1]$$ [ - 1 , 1 ] that is analytic in a Bernstein ellipse with parameter $$\rho > 1$$ ρ > 1 , and for a uniform perturbation level $$\varepsilon $$ ε on the function samples, we construct an asymptotically best extrapolant e(x) as a least squares polynomial approximant of degree $$M^*$$ M ∗ determined explicitly. We show that the extrapolant e(x) converges to f(x) pointwise in the interval $$I_\rho \in [1,(\rho +\rho ^{-1})/2)$$ I ρ ∈ [ 1 , ( ρ + ρ - 1 ) / 2 ) as $$\varepsilon \rightarrow 0$$ ε → 0 , at a rate given by a x-dependent fractional power of $$\varepsilon $$ ε . More precisely, for each $$x \in I_{\rho }$$ x ∈ I ρ we have $$\begin{aligned} |f(x) - e(x)| = \mathcal {O}\left( \varepsilon ^{-\log r(x) / \log \rho } \right) , \quad r(x) = \frac{x+\sqrt{x^2-1}}{\rho }, \end{aligned}$$ | f ( x ) - e ( x ) | = O ε - log r ( x ) / log ρ , r ( x ) = x + x 2 - 1 ρ , up to log factors, provided that an oversampling conditioning is satisfied, viz. $$\begin{aligned} M^* \le \frac{1}{2} \sqrt{N}, \end{aligned}$$ M ∗ ≤ 1 2 N , which is known to be needed from approximation theory. In short, extrapolation enjoys a weak form of stability, up to a fraction of the characteristic smoothness length. The number of function samples does not bear on the size of the extrapolation error provided that it obeys the oversampling condition. We also show that one cannot construct an asymptotically more accurate extrapolant from equally spaced samples than e(x), using any other linear or nonlinear procedure. The proofs involve original statements on the stability of polynomial approximation in the Chebyshev basis from equally spaced samples and these are expected to be of independent interest. 2021-09-20T17:17:21Z 2021-09-20T17:17:21Z 2018-03-21 2020-09-24T21:23:01Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131505 en https://doi.org/10.1007/s10208-018-9384-1 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ SFoCM application/pdf Springer US Springer US |
| spellingShingle | Demanet, Laurent Townsend, Alex Stable Extrapolation of Analytic Functions |
| title | Stable Extrapolation of Analytic Functions |
| title_full | Stable Extrapolation of Analytic Functions |
| title_fullStr | Stable Extrapolation of Analytic Functions |
| title_full_unstemmed | Stable Extrapolation of Analytic Functions |
| title_short | Stable Extrapolation of Analytic Functions |
| title_sort | stable extrapolation of analytic functions |
| url | https://hdl.handle.net/1721.1/131505 |
| work_keys_str_mv | AT demanetlaurent stableextrapolationofanalyticfunctions AT townsendalex stableextrapolationofanalyticfunctions |