...c'' if for any $x_0\in I$ there is a neighborhood $J$ of $x_0$ and a power series $\sum a_n (x-x_0)^n$ such that
...write an identity as \eqref{e:power_series} we implicitly assume that the series on the left hand side converges). The notion can be extended to functions
6 KB (1,048 words) - 21:19, 14 January 2021
$#C+1 = 41 : ~/encyclopedia/old_files/data/N066/N.0606440 Neumann series
A series of the form
6 KB (896 words) - 19:13, 17 January 2024
the prediction is called linear if one restricts consideration to linear estimators).
One of the problems posed and solved was that of linear prediction of a stationary sequence. This problem is analogous to the follo
6 KB (908 words) - 08:23, 6 June 2020
A general theory of linear integral equations of the second kind,
established by D. Hilbert [[#References|[1]]] on the basis of his theory of linear and bilinear forms in an infinite number of variables. The principal idea o
3 KB (478 words) - 22:10, 5 June 2020
there exists a bounded linear operator $ \delta f ( a , \cdot ) $
if for each continuous linear functional $ y ^ \prime $
6 KB (901 words) - 16:08, 1 April 2020
...er of the series $F$, and the form $F_k$ is called the initial form of the series.
are two formal power series, then, by definition,
6 KB (1,093 words) - 08:26, 16 March 2023
...notion of a [[Limit|limit]] is defined. Here are the simplest examples of series.
==Simple series of numbers.==
29 KB (4,393 words) - 19:21, 27 January 2020
...nt groups and extensions. The number of infinite factors in any polycyclic series is an invariant of the polycyclic group (the polycyclic dimension). The hol
Every solvable linear group over the integers is polycyclic, [[#References|[a1]]]. A solvable gro
4 KB (588 words) - 19:59, 11 April 2014
The Euler transformation of series. Given a series
the series
6 KB (972 words) - 12:59, 6 January 2024
are the partial sums of the [[Fourier series|Fourier series]] of a function $ f $
...ral properties which are of interest in the theory of summation of Fourier series. For instance, if $ p = [ cn] $,
6 KB (901 words) - 17:32, 5 June 2020
...only if it can be reduced to triangular form over $k$. If $k = 0$, then a linear algebraic group defined over $k$ is split over $k$ if and only if its Lie a
...</td></tr><tr><td valign="top">[4]</td> <td valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </td>
4 KB (592 words) - 13:44, 17 October 2019
...rential equations may be employed to find approximate solutions of the non-linear systems.
...ed by a special method, known as harmonic linearization. Let the given non-linear function (force) be
4 KB (613 words) - 19:43, 5 June 2020
A [[Linear algebraic group|linear algebraic group]] defined over $ k $
Here a connected solvable linear algebraic group $ B $
4 KB (559 words) - 16:39, 17 December 2019
...series in a Banach space is weakly subseries convergent if and only if the series is norm subseries convergent; this result was announced in [[#References|[a
...eries in $X$ is $\sigma ( X , Y )$ subseries convergent if and only if the series is norm subseries convergent. If $X$ is the dual of a Banach space $Z$ and
5 KB (714 words) - 15:30, 1 July 2020
...for studying certain types of sequences of polynomials, or formal Laurent series, through the use of modern algebra techniques.
...space. In fact, $P^*$ is isomorphic to the vector space of [[formal power series]] $\mathcal{F}$ via the mapping
7 KB (1,142 words) - 20:34, 9 December 2015
is linear and continuous from $ \mathcal D ( O) $
is linear and continuous from $ \mathcal D ^ \prime ( O) $
4 KB (499 words) - 08:03, 25 April 2022
and this series converges in $ L _ {p ^ \prime } $.
...n="top"> S.G. Krein, Yu.I. Petunin, E.M. Semenov, "Interpolation of linear operators" , Amer. Math. Soc. (1982) (Translated from Russian)</TD></TR><
4 KB (525 words) - 22:10, 5 June 2020
...earer formulation is possible for certain theorems on the convergence of a series
...ype of theorem is the [[Riesz-Fischer theorem|Riesz–Fischer theorem]]: The series
2 KB (321 words) - 18:58, 7 September 2017
...[[Series expansion|Series expansion]]), which can be used in the study of linear elliptic operators, see [[#References|[a1]]]. A general reference for Carle
2 KB (299 words) - 09:13, 26 March 2023
===Taylor series of a smooth function===
If $f$ is infinitely smooth near the origin $x=0$, then its Taylor series (at the origin) has the form
3 KB (512 words) - 06:16, 13 June 2022