...}^\infty a_k$ is summable by the summation method to the sum $A$, then the series $\sum_{k=0}^\infty ca_k$ is summable by this method to the sum $cA$;
...are summable by the summation method to $A$ and $B$ respectively, then the series $\sum_{k=0}^\infty(a_k+b_k)$ is summable by this method to the sum $A+B$.
2 KB (276 words) - 20:05, 9 November 2014
...when the sequence of its terms is arbitrarily rearranged. More exactly, a series
of elements of a linear space $ E $
3 KB (411 words) - 12:53, 6 January 2024
...ts, X_{n-m}$. A linear auto-regression scheme of order $m$ is defined by a linear [[Regression|regression]] equation between $X_n$ and $X_{n-k}$, $k=1,\ldots
...roduced by [[Yule, George Udny|G. Yule]] in 1921) in order to analyze time series describing a system which is oscillating under the effect of internal force
1 KB (180 words) - 01:22, 15 February 2024
...representations connected with the complementary series of adèle groups of linear algebraic groups.
<TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Naimark, "Linear representations of the Lorentz group" , Macmillan (1964) (Translated from R
3 KB (393 words) - 18:23, 26 October 2014
Summability of a series for all possible rearrangements of its terms. The series
...\infty|\eta_n|<\infty$; moreover, $c=0$ if the method $A$ does not sum the series $\sum_{n=1}^\infty1$ [[#References|[3]]].
3 KB (434 words) - 15:11, 14 February 2020
A [[series]] of [[Real number|real]] or [[Complex number|complex numbers]], or more in
for which the series of real numbers
5 KB (821 words) - 09:35, 16 August 2013
Problems in which optimal series of products and their component parts have to be determined.
...rse of the development, production and use of all the products. An optimal series of products exists when, as the number of distinct types of products increa
5 KB (732 words) - 06:42, 15 August 2014
...ries and sequences of numbers; established by K. Weierstrass . If, for the series
there exists a convergent series of non-negative numbers
3 KB (407 words) - 20:50, 19 December 2020
$#C+1 = 33 : ~/encyclopedia/old_files/data/V096/V.0906870 Volterra series,
''integro-power series''
5 KB (717 words) - 08:28, 6 June 2020
$#C+1 = 53 : ~/encyclopedia/old_files/data/D033/D.0303980 Double series
A series
7 KB (1,082 words) - 19:36, 5 June 2020
...n" , that is, if the class contains every group having an ascending normal series with factors from the given class (see
[[Normal series|Normal series]]). In every group there is a largest radical normal subgroup — the radic
3 KB (517 words) - 11:55, 23 November 2013
...he main type of integral equations suitable for solving by this method are linear one-dimensional integral Fredholm equations of the second kind. The method
Solving (2) is reduced to solving a system of linear algebraic equations. The degenerate kernel $ K _ {N} ( x, s) $
3 KB (401 words) - 17:32, 5 June 2020
...Maclaurin series. The series was published by B. Taylor in 1715, whereas a series reducible to it by a simple transformation was published by Johann I. Bern
...rentiable at $x_0$, its Taylor series at $x_0$ is the [[Power series|power series]] given by
4 KB (710 words) - 06:13, 13 June 2022
A non-linear integral equation of the form
...n integro-power series of two functional arguments. It is assumed that the series converges absolutely and uniformly.
4 KB (524 words) - 04:11, 6 June 2020
series''
A series of functions used in the representation of automorphic forms and functions
5 KB (778 words) - 08:25, 6 June 2020
...em $\{\psi_n\}$ is orthonormal, $g_n=\psi_n$ for all $n\in\mathbf N$. If a series
...si_n\}$. In the proofs of a number of theorems in the theory of orthogonal series, the [[Bessel inequality|Bessel inequality]] and the [[Riesz–Fischer theo
4 KB (643 words) - 19:53, 27 February 2021
A bounded linear [[Integral operator|integral operator]] $ T $
The series (1) converges with respect to the norm of $ L _ {2} ( X, \mu ) $,
5 KB (758 words) - 22:10, 5 June 2020
...en recurrence relation has an analogy with solving an ordinary homogeneous linear differential equation with constant coefficients.
..._{n-1}+\dotsb+p_m\alpha_{n-m}$, $n\geq m$, if and only if the formal power series $\alpha(x)=\alpha_0+\alpha_1x+\dotsb$ is a rational function of the form $\
1 KB (226 words) - 13:04, 14 February 2020
in the Taylor–Fourier series
satisfy a linear relation as follows:
4 KB (630 words) - 08:11, 6 June 2020
...eries|Fourier series]]; [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]], and also expansions in general orthogonal syst
...d Faddeev are difficult to verify for concrete singular integrals, a whole series of papers was devoted to the search for effective sufficient conditions for
6 KB (837 words) - 08:14, 6 June 2020