...terion first proved by Lebesgue for the convergence of Fourier series in {{Cite|Le}}.
...rion|Young criterion]]. Cp. with Section 7 of Chapter III in volume 1 of {{Cite|Ba}}.
2 KB (235 words) - 11:59, 14 December 2012
...ere is a generalization of the Prüfer surface to any even dimension (cf. {{Cite|2}}).
857 bytes (113 words) - 15:05, 30 March 2023
An inequality proved by Simons in his fundamental work {{Cite|Si}} on minimal varieties, which played a pivotal role in the solution of t
...e special case of $2$-dimensional minimal surfaces of $\mathbb R^3$ (cf. {{Cite|CM}}).
3 KB (425 words) - 21:27, 13 October 2014
...e}^z$ at the essential singularity $a=\infty$. This result of F. Iversen {{Cite|Iv}} supplements the big [[Picard theorem]] on the behaviour of an analytic
...harmonic functions on $\R^n$, notably by W.K. Hayman, see {{Cite|HaKe}}, {{Cite|Ha}}.
1 KB (198 words) - 21:07, 31 July 2012
...{{Cite|AlUr}}) was established in {{Cite|Sm}} for $\tau=\aleph_0$ and in {{Cite|Ar}} in complete generality. Cf. [[Weight of a topological space|Weight of
1 KB (203 words) - 15:05, 1 May 2014
...therwise $E$ is said to be of the second category (cp. with Chapter 9 of {{Cite|Ox}}).
...ame for such sets is residual (or comeager), cp. again with Chapter 9 of {{Cite|Ox}}. A non-empty closed set of real numbers, in particular an interval, is
2 KB (291 words) - 19:06, 7 December 2023
...Cite|N}}); its analogue for continuous time has since been obtained (see {{Cite|AC}}).
...rem, which is itself subject to various generalizations (see {{Cite|C}}, {{Cite|T}}, as well as the bibliography under [[Operator ergodic theorem|Operator
3 KB (445 words) - 22:08, 7 July 2014
{{Cite|CaFr}}, and so it is valid for algebraic curves of genus
{{Cite|Ca}}). For quadrics over a number field the Hasse
3 KB (543 words) - 15:48, 17 February 2012
A criterion first proved by Dini for the convergence of Fourier series in {{Cite|Di}}.
...w0}[f(x+t)+f(x-t)]$ exists, then $S=\lim_{t\downarrow0}[f(x+t)+f(x-t)]$. {{Cite|Di}}
2 KB (346 words) - 13:51, 27 June 2017
...ewise monotone functions, is also called Dirichlet-Jordan test, cf. with {{Cite|Zy}}.
For a proof see Section 10.1 and Exercises 10.13 and 10.14 of {{Cite|Ed}}.
1 KB (205 words) - 12:30, 27 September 2012
...y countable family of dense open subsets is dense (cp. with Section 9 of {{Cite|Ox}}
...] [[Hausdorff space|Hausdorff spaces]] (see Section 5.3 of Chapter IX in {{Cite|Bo}}).
2 KB (353 words) - 17:21, 18 August 2012
...m algorithms, such as the Cooley–Tukey fast Fourier transform algorithms {{Cite|CoTu}}, depend on the existence of non-trivial divisors of the transform si
...$p$ could be computed by a $(p-1)$-point cyclic convolution. S. Winograd {{Cite|Wi}} extended Rader's result to Fourier transform computations of prime pow
4 KB (535 words) - 22:23, 25 April 2012
(See {{Cite|C|Sect. 8.4}}, {{Cite|S|p. 170}}.)
...σ-algebra]], is treated as a measurable space. (See {{Cite|S|p. 170}}, {{Cite|D|Sect. 11.5}}.)
5 KB (655 words) - 20:33, 18 February 2012
{{Cite|Se}}.
{{Cite|Se}}. Furthermore, under this condition ${\rm Ext}_\cO^p(\cF,\cS)$ is coher
3 KB (497 words) - 13:34, 6 January 2022
{{Cite|Sc2}},
{{Cite|Bo}}, 14.12, of $G/P$ (cf. also
3 KB (371 words) - 18:45, 30 March 2012
The series (*) were introduced by F.Y. Edgeworth {{Cite|E}}. Their asymptotic properties have been studied by H. Cramér, who has s
...15 years (as of 1988). An important recent contribution in this area is {{Cite|BGZ}}.
6 KB (866 words) - 11:09, 12 May 2012
...ined in the following way (see, for example, Exercise 46 in Chapter 2 of {{Cite|Ro}}). Given $x\in [0,1]$ consider its ternary expansion $\{a_i\}$, i.e. a
...alternative definitions see Example 1.67 of {{Cite|AFP}} and page 55 of {{Cite|Co}}.
2 KB (369 words) - 11:26, 13 December 2014
...ion first proved by W. H. Young for the convergence of Fourier series in {{Cite|Yo}}.
Cp. with Section 4 of Chapter III in volume 1 of {{Cite|Ba}}.
2 KB (233 words) - 20:45, 16 October 2012
...kind). These equations arose in the context of the studies of N.H. Abel {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of
...t be integrated in closed form, though this is possible in special cases {{Cite|Ka}}. If $g_0,g_1 \in C^1(a,b)$ and $g_1(x) \neq 0$, $g_0(x) + g_1(x)y \neq
2 KB (338 words) - 22:19, 4 July 2012
...ization of measurability (Stepanov–Denjoy theorem, see Theorem 2.9.13 of {{Cite|Fe}}).
(cp. with Section 1.7.2 of {{Cite|EG}}). Conversely, if $f$ is essentially bounded, the points of approximate
3 KB (373 words) - 17:15, 18 August 2012