An element $e$ in a set $M$ is called a left (right) unit (left (right) identity) with respect to a [[binary operation]] $*$ defined
...unit is used for only one of these operations, usually multiplication. The unit with respect to addition is called the [[zero]] element.
2 KB (403 words) - 19:25, 3 April 2016
...m''' asks how large the values of such a polynomial must be on the [[unit circle]] in the [[complex plane]]. The answer to this would yield information ab
...= \pm 1$. Let $\Vert p \Vert$ denote the supremum of $p(z)$ on the unit circle. ''Littlewood's problem'' asks for constants $c_1$ and $c_2$ such that th
1 KB (187 words) - 21:08, 23 November 2023
...le has 180 degrees. Degrees are also used to measure circular arcs (a full circle has 360 degrees).
...ee of irrationality, and many more. It is also, of course, the name of the unit for temperature measurements in various scales.
911 bytes (147 words) - 19:34, 27 December 2014
...$ radians; it is approximately $57^\circ17'44''$. A radian is taken as the unit of measurement of angles in the so-called circular, or radian, measurement
464 bytes (79 words) - 21:25, 11 April 2014
...e real axis (the equation is stable), then the multipliers lie on the unit circle. Consider a canonical equation $\dot x = i \lambda J H(t) x$ with a real pa
...ability it is necessary and sufficient that all multipliers be on the unit circle and that there be no coincident multipliers of different kinds.
2 KB (348 words) - 19:54, 26 November 2016
then the circle $\lvert z\rvert=R$ is a [[natural boundary]]: all points of the cicle are s
...t sequence $p_n$, radius of convergence equal to 1, but for which the unit circle is not a natural boundary.
2 KB (290 words) - 18:23, 10 October 2023
Szegö quadrature formulas are the analogues on the unit circle $\bf T$ in the complex plane of the Gauss quadrature formulas on an interva
...i_n$ as nodes (as in Gaussian formulas), because these are all in the open unit disc $\mathbf D$ (cf. also [[Szegö polynomial|Szegö polynomial]]). Theref
3 KB (454 words) - 16:59, 1 July 2020
...ert kernel and the [[Cauchy kernel|Cauchy kernel]] in the case of the unit circle:
902 bytes (131 words) - 20:27, 18 March 2024
...[[Multipliers|Multipliers]]) of the variational equation lies on the unit circle. There are also results about the local structural stability of certain hyp
3 KB (433 words) - 12:20, 26 July 2014
...nces|[4]]]). There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [[#Referenc
2 KB (271 words) - 14:01, 1 October 2014
...al power dilation with spectrum in $\partial S$. The minimal radius of the circle which is a spectral set for every contraction in a Banach space is equal to
...> J. von Neumann, "Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes" ''Math. Nachr.'' , '''4''' (1951) pp. 258–281</TD></TR><T
2 KB (295 words) - 15:46, 29 December 2018
...a closed curve $L'$ with a cusp at the point $w=1$, touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. ban
The function $w=\lambda(\rho t+\alpha)$ maps the exterior of the unit circle in the $t$-plane to the exterior of $L'$. To obtain a Zhukovskii profile of
4 KB (542 words) - 19:58, 4 January 2024
be a bounded regular analytic function in the unit disc $ \Omega = \{ {z \in \mathbf C } : {| z | < 1 } \} $,
...s serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and
4 KB (546 words) - 08:06, 6 June 2020
...omeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.
1 KB (182 words) - 19:25, 13 December 2015
...n a so-called great circle is obtained as the intersection. A unique great circle can be drawn through any two points $A$ and $B$ on the sphere (Fig. a), exc
...a straight line is the shortest curve between its ends, an arc of a great circle on a sphere is only the shortest curve when it is shorter than the compleme
8 KB (1,389 words) - 15:53, 19 April 2014
...fmath.org/legacyimages/a/a013/a013980/a0139808.png" /> lie inside the unit circle, then equation (*) has the solution
5 KB (730 words) - 17:11, 7 February 2011
...ally onto a standard pair $(D^n,D^m)$ or $(D^n,D_+^m)$, where $D^k$ is the unit ball of the space $\mathbf R^k$ with centre at the origin and $D_+^k$ is th
...imbedding of a circle and an arc into a plane is locally flat; however, a circle or an arc can be imbedded in $\mathbf R^k$ with $k\geq3$ in a manner that i
2 KB (383 words) - 08:32, 19 April 2014
...nd bijectively. The circle property: Under a fractional-linear mapping any circle in $ \overline{\mathbf C}\; $(
i.e. a circle in $ \mathbf C $
13 KB (1,875 words) - 13:58, 17 March 2023
...axis of revolution and the circle described by the centre of its rotating circle.
...f $u$ and $v$, $r$ is the radius vector of the surface $F$, and $n$ is the unit normal to $F$.
2 KB (440 words) - 16:55, 3 August 2014
in the open unit disc whose $ H ^ \infty $-
...hur algorithm can also be used to obtain a Routh or Jury test for the open unit disc, that is, the Schur algorithm can be used to determine whether or not
6 KB (836 words) - 11:17, 30 May 2020
is a bounded regular [[Analytic function|analytic function]] in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
of the circle $ \Gamma = \{ {z } : {| z | = 1 } \} $
5 KB (791 words) - 08:11, 6 June 2020
of measure zero on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $,
that is regular, analytic and bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $
3 KB (354 words) - 04:11, 6 June 2020
...p"> P.E. Blanksby, H.L. Montgomery, "Algebraic integers near the unit circle" ''Acta Arith.'' , '''18''' (1971) pp. 355–369</TD></TR>
7 KB (1,029 words) - 07:50, 27 March 2018
...o, then $f(z)=0$ in $D$. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set $E$ of the given t
3 KB (424 words) - 21:56, 24 July 2012
...mages/c/c026/c026230/c0262306.png" /> rotates only in one direction as the circle is traversed. The following inequality expresses a necessary and sufficient
....org/legacyimages/c/c026/c026230/c02623054.png" />, and the radius of this circle cannot be increased without imposing additional restrictions on the class o
19 KB (2,650 words) - 17:07, 7 February 2011
...nction $u(z)$, $z=r\mathrm{e}^{\mathrm{i}\phi}$, can be represented in the unit disc $U=\{ z\in\C : \abs{z} < 1 \}$ by a Poisson–Stieltjes integral
where $\mu$ is a Borel measure concentrated on the unit circle $T=\{ z\in\C : \abs{z} = 1 \}$, $\int\rd\mu(\xi)=1$. Then almost-everywhere
4 KB (706 words) - 19:19, 27 July 2012
in the unit disc $ D $
on the boundary circle $ C $(
5 KB (698 words) - 18:20, 26 January 2022
...n|holomorphic functions]] of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kernel and the [[Hilb
1 KB (198 words) - 20:26, 18 March 2024
...nt lifting theorem and a certain contractive analytic function in the open unit disc. This characterization of all solutions has several different network
on the unit circle whose norm $ \| g \| _ \infty = { \mathop{\rm ess} \sup } \{ {| {g ( e
6 KB (815 words) - 09:51, 26 March 2023
inside the unit circle on the complex plane. Similarly, the dynamical
polynomial $w(z)$ has $m$ roots inside and $n-m$ roots outside the unit
4 KB (607 words) - 02:33, 14 September 2022
lies on the unit circle and there is the spectral decomposition $ U = \int _ {0} ^ {2 \pi } e ^ {
2 KB (255 words) - 08:27, 6 June 2020
on the open unit disc $ | z | < 1 $
...I. Schur, "On power series which are bounded in the interior of the unit circle. II" I. Gohberg (ed.) , ''Methods in Operator Theory and Signal Processing
5 KB (791 words) - 05:51, 13 June 2022
under which the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $
is mapped onto itself, the maximum diameter of the image of the circle $ \Gamma _ {R} = \{ {z } : {| z | = R } \} $
4 KB (623 words) - 19:42, 5 June 2020
the circle $ | z | = 1 $
A hyperbolic circle in $ D $
7 KB (952 words) - 12:50, 13 January 2024
where $z\rightarrow 1$ along any path not tangent to the unit circle.
2 KB (258 words) - 12:23, 10 January 2015
...ta$ are real, namely $\theta$ and $1/\theta$, and the rest lie on the unit circle. The field $\mathbf{Q}(\theta)$ is thus a quadratic extension (cf. [[Extens
3 KB (504 words) - 18:43, 14 April 2023
...$ can be characterized as the space of analytic functions $f ( z )$ on the unit disc having [[Taylor series|Taylor series]] representation
...ions (and, more generally, of contractive analytic matrix-functions on the unit disc) in terms of Schur parameters (see [[#References|[a9]]] and [[#Referen
12 KB (1,802 words) - 17:01, 1 July 2020
be a meromorphic function in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
on the circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $
4 KB (660 words) - 08:06, 6 June 2020
Along with these, there are examples of bounded analytic functions in the unit disc $ D $
of measure zero on the unit circle $ \Gamma $.
10 KB (1,496 words) - 08:27, 6 June 2020
The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real
==Orthogonal polynomials on the circle.==
9 KB (1,315 words) - 20:15, 12 January 2024
...e unit disc $D = \{ z \in \mathbf{C} : |z| < 1 \}$; then all points of the circle $\Gamma = \{ z \in \mathbf{C} : |z| = 1 \}$ except, possibly, for a [[First
2 KB (313 words) - 19:43, 18 April 2017
...aofmath.org/legacyimages/m/m064/m064430/m06443021.png" /> is an arc of the
circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...ar group (1) is then replaced by the modular group of automorphisms of the
unit disc. For example, it is convenient to apply the fractional-linear transfor
39 KB (5,287 words) - 17:07, 7 February 2011
...explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. [[Geometric constructions|Geometric construc
4 KB (680 words) - 13:40, 30 December 2018
of functions of bounded type in the unit disc $ \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $:
almost-everywhere on the unit circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $;
8 KB (1,170 words) - 19:40, 5 June 2020
...(cf. [[Spectrum of an operator|Spectrum of an operator]]) lies on the unit circle, and $U$ has a representation
2 KB (433 words) - 12:44, 18 August 2014
analytic on the open unit disc $ | z | < 1 $(
...s the [[Banach algebra|Banach algebra]] of complex-valued functions on the unit disc having a [[Fourier series|Fourier series]]
6 KB (869 words) - 11:06, 30 May 2020
...opediaofmath.org/legacyimages/b/b110/b110130/b11013061.png" /> on the
unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...w.encyclopediaofmath.org/legacyimages/b/b110/b110130/b11013075.png" /> and
unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
41 KB (5,422 words) - 22:26, 1 January 2018
...ametrized by a unit vector $\alpha \in S ^ { 1 }$, $S ^ { 1 }$ is the unit circle in $\mathbf{R} ^ { 2 }$ and $p \in \mathbf R _ { + } : = [ 0 , \infty )$. B
If $n_0$ is a unit vector normal to $S$ at the point $x _ { 0 }$, then for an arbitrary $\gamm
6 KB (922 words) - 14:50, 27 January 2024
Let $a$ be a complex-valued function defined on the complex unit circle $\bf T$, with [[Fourier coefficients|Fourier coefficients]]
...he exponentials of continuous complex-valued functions defined on the unit circle.
13 KB (1,838 words) - 07:22, 13 February 2024
is the unit $ n \times n $
...style="background-color:white;" colspan="1">Situated inside or on the unit circle; in the latter case simple elementary divisors of the monodromy matrix corr
22 KB (3,165 words) - 08:59, 21 January 2024
.../h/h046/h046320/h0463206.png" /> is the normalized Lebesgue measure on the
circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...046/h046320/h046320134.png" /> are defined by the condition (*), where the
circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
37 KB (5,073 words) - 18:20, 1 December 2014
At points of the unit circle $ | z | = 1 $
3 KB (470 words) - 08:17, 26 March 2023
the convergence also extends over the unit circle if $ \mathop{\rm Re} ( \alpha + \beta - \gamma ) < 0 $;
it converges at all points of the unit circle except $ z = 1 $.
12 KB (1,576 words) - 01:14, 21 January 2022
onto the unit disc $ | \zeta | \leq 1 $
there is a one-to-one correspondence between the points of the circle and the prime ends of $ B $,
5 KB (847 words) - 22:16, 5 June 2020
...op"> L. Carleson, "Sets of uniqueness for functions regular in the unit circle" ''Acta Math.'' , '''87''' : 3–4 (1952) pp. 325–345</TD></TR><TR><T
3 KB (458 words) - 10:23, 2 June 2020
to be normal in the unit disc $ G = \{ {z \in \mathbf C } : {| z | < 1 } \} $
in the unit disc $ G $
7 KB (1,050 words) - 08:03, 6 June 2020
having equal multiplicity. Roots of unit modulus have even multiplicity. It follows that $ w ( z ) = c \prod _ {1}
on the unit circle $ \partial D = \{ {e ^ {it } } : {0 \leq t < 2 \pi } \} $.
9 KB (1,334 words) - 13:15, 26 March 2023
Let $\Delta$ be the open unit disc in the complex plane $\mathbf{C}$, and let $\operatorname{Hol}( \Delta
For $\omega$ on the unit circle $\partial \Delta$, the boundary of $\Delta$, and $\alpha > 1$, a non-tangen
14 KB (2,118 words) - 16:09, 11 February 2024
A circle of unit radius with diametrically-opposite points identified in the Euclidean plane
...elliptic plane is homeomorphic to the real projective plane). A sphere of unit radius with antipoles identified in three-dimensional Euclidean space can s
19 KB (3,123 words) - 23:52, 14 December 2020
denotes the closed unit disc and $ A ( D ) $
the unit circle) such that $ | {f ( t ) } | = \| f \| _ \infty $.
8 KB (1,167 words) - 11:37, 12 January 2021
self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values $ \lam
...the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for $ J $-
9 KB (1,299 words) - 08:07, 6 June 2020
and, in particular, by the unit circle.
...is the theorem of Fatou (1906): If an analytic function is bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $,
25 KB (3,676 words) - 08:39, 26 March 2023
...$A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, the measure
7 KB (1,114 words) - 19:36, 23 December 2023
...gle is the length of the arc cut out by the sides of the angle on the unit circle. One radian is assigned to the angle the arc of which has length equal to t
6 KB (1,009 words) - 11:05, 4 December 2016
...contractive operator $T$ in $D$, together with the complement in the unit circle of the union of the arcs along which $m _ { T } ( \lambda )$ can be analyti
6 KB (897 words) - 20:02, 27 February 2021
...and only if there exists a function $\phi \in L ^ { \infty }$ on the unit circle $\bf T$ such that
...t{o}$ and $\text{o}$ is a [[Conformal mapping|conformal mapping]] from the unit disc onto the upper half-plane.
14 KB (2,017 words) - 06:44, 15 February 2024
...ons of the space $\mathbb{R}^3$ around the origin and also to the space of unit tangent vectors to the sphere $S^2$; 4) all compact zero-dimensional groups
...of them (compactness, connectedness, etc.; e.g., a segment differs from a circle in that it can be divided into two by one point); the method of invariants
11 KB (1,634 words) - 17:03, 23 March 2023
...connection between the boundary properties of an analytic function in the unit disc and the metric of the Riemann surface onto which it maps the disc (see
be any domain inside the unit disc $ D= \{ {z } : {| z | < 1 } \} $
7 KB (1,075 words) - 19:29, 1 January 2021
...ction|Generating function]]) appeared as the source of the [[Circle method|circle method]] of Hardy–Littlewood–Ramanujan and the method of trigonometric
...raightedge and a compass (cf. [[Quadrature of the circle|Quadrature of the circle]]) led to questions about the arithmetic nature of the number $ \pi $,
10 KB (1,503 words) - 08:03, 6 June 2020
...g]] of a triangle bounded by arcs of circles onto the upper half-plane (or unit disc) that remains single-valued under unrestricted [[Analytic continuation
...warz functions are automorphic functions with a continuous singular curve (circle or straight line). In particular, the cases of $ \nu _ {1} = 2 $,
5 KB (568 words) - 08:12, 6 June 2020
...cative linear functional has norm 1, each such a functional belongs to the unit sphere of the dual of $ A $.
is closed in the weak topology on the dual space. Since the unit ball is compact in the weak topology on the dual space, $ \Phi $
18 KB (2,806 words) - 03:47, 25 February 2022
...Circular arcs (and diameters) in $E$ which are orthogonal to the boundary circle $\Omega=\{z\colon|z|=1\}$ are called hyperbolic lines. Every point of $\Ome
5 KB (705 words) - 19:16, 14 August 2014
...mbrane is circular. In other words, among all membranes of given area, the circle has the lowest fundamental frequency. This inequality was conjectured by Lo
...i ^ { n / 2 } / \Gamma ( n / 2 + 1 )$ is the volume of the $n$-dimensional unit ball. Equality is attained in (a5) if and only if $\Omega$ is a ball.
13 KB (1,910 words) - 18:40, 19 February 2024
is the unit disc. Then $ G $
be a Fuchsian group acting on the unit disc $ D $.
8 KB (1,210 words) - 10:58, 29 May 2020
has a unit which is the $ \delta $-
of zeros and ones is the Fourier–Stieltjes transform of some measure on the circle if and only if $ (c _ {m} ) $
4 KB (655 words) - 13:07, 7 April 2023
...ubdomain of $B ( 0,1 ) \subseteq \mathbf C$, bounded by an arc of the unit circle and a smooth simple curve $\Gamma \subseteq B ( 0,1 )$ and assume that $f \
...2]]] for references) in the bounded "version" of $H ^ { n }$, namely the unit ball $B$ of $\mathbf{C} ^ { n + 1}$, or, more generally, for bounded domai
6 KB (961 words) - 16:45, 1 July 2020
...rnel]] and as the [[Cauchy kernel|Cauchy kernel]]. In the case of the unit circle, there exists a simple relationship between these kernels:
5 KB (696 words) - 22:17, 28 January 2020
and the selected scale unit $ e $,
...of the number of solutions of the set of equations of the line and of the circle.
8 KB (1,255 words) - 18:47, 5 April 2020
...n over all $H \in H ^ { 2 } ( \mu , {\bf D} )$ ($\mathbf D$ being the open unit disc) satisfying $H ( 0 ) = 1$. If $H$ is restricted to be a polynomial of
...Carathéodory or positive real function because it is analytic in the open unit disc and has positive real part there.
7 KB (1,105 words) - 10:02, 11 November 2023
...464201.png" /> be a complex-valued square-summable function on a
circle of
unit length (or on the segment <img align="absmiddle" border="0" src="https://ww
...al form for results of classical harmonic analysis on the real line or the
circle, but also establishes new results regarding larger classes of topological g
66 KB (9,085 words) - 17:28, 31 March 2020
is the unit circle and $ f _ {i} : S _ {i} \rightarrow S _ {i-} 1 $
4 KB (631 words) - 08:14, 6 June 2020
...polynomials asks how large the values a polynomial must be on the [[unit circle]] in the [[complex plane]] when the coefficients of the polynomial are all
4 KB (641 words) - 08:31, 23 November 2023
algebra is said to be an algebra with a unit if $A$ contains an element $e$ such that $ex=xe=x$ for any
$x\in A$. If a Banach algebra has no unit, a unit may be adjoined, i.e. it is possible to construct a
14 KB (2,346 words) - 22:48, 29 November 2014
...0$ and the roots of $P ( x )$ other than $\theta$ all lie in the open unit circle $| x | < 1$. The set of these numbers is traditionally denoted by $S$. Ever
5 KB (723 words) - 19:03, 23 January 2024
which is homeomorphic to the circle $ S ^ {1} $(
the unit sphere in the Euclidean space $ \mathbf R ^ {3} $).
6 KB (797 words) - 22:12, 5 June 2020
is the parameter on the unit circle $ S ^ {1} $,
...d, D.B. Fuks, "The cohomology of the Lie algebra of vector fields in a circle" ''Funct. Anal. Appl.'' , '''2''' (1968) pp. 342–343 ''Funkts. Anal.
10 KB (1,539 words) - 11:16, 23 March 2023
...s in the upper half-plane, and that of a unitary operator lies on the unit circle). If the scalar product is not of fixed sign, but its index of indefinitene
points outside the unit circle. For $ J $-
16 KB (2,424 words) - 08:22, 6 June 2020
...logarithmic Mahler measure $m ( P )$ is defined to be the average over the unit $n$-torus of $\operatorname { log } | P ( x _ { 1 } , \dots , x _ { n } ) |
...lign="top"> C.J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer" ''Bull. London Math. Soc.'' , '''3''' (1971) pp. 1
7 KB (1,101 words) - 14:50, 27 January 2024
...refers to representations of an invertible matrix-function $f$ on the unit circle $\mathcal{T}$ of the form $f = f_+ . \delta . f_-$, where $f _ { \pm }$ are
...standard decomposition of the Riemann sphere $S$, where $D _ { + }$ is the unit disc and $D_{-}$ is the complementary domain containing the point $\{ \inft
7 KB (989 words) - 16:56, 1 July 2020
...ecial case of a normal operator, the spectral measure is given on the unit circle. The spectral decomposition of a unitary operator $ U $
5 KB (793 words) - 08:22, 6 June 2020
...Toeplitz operator]]) whose symbol $f$ is a continuous function on the unit circle. In this case $\sigma _ { \text { lre } } ( T )$ is the range of $f$, and $
5 KB (812 words) - 17:00, 1 July 2020
...the second through the smaller angle is the direction of traversal on the circle. In $E^3$, a frame can be related to a screw. The first vector goes in the
...tangent plane at the point $x_0$ (for example, a rotation direction on the circle can be defined by choosing one tangent vector). If $M$ has a boundary and i
18 KB (2,980 words) - 17:05, 13 June 2020
...s to consideration of examples (a straight line, an interval, a polygon, a circle, etc.). Since it does not have general methods at its disposal, elementary
...ex of each end of the interval is equal to one. b) At each of its points a circle has ramification index two. c) A curve consisting of $ n $
19 KB (2,971 words) - 19:30, 26 March 2023
...algebra|Lie algebra]] consisting of the differential operators on the unit circle $S ^ { 1 }$ which have a finite Fourier expansion (cf. also [[Fourier serie
...ghest weight modules over the Lie algebra of differential operators on the circle" ''Comm. Math. Phys.'' , '''157''' (1993) pp. 429–457</td></tr><tr><td
9 KB (1,398 words) - 10:21, 11 November 2023
...genus, Willmore also showed that for anchor rings, obtained by rotating a circle of radius $r$ about an axis in its plane at distance $R > r$ from its ce
...nding Euler equation. In the particular case when the ambient space is the unit sphere $S ^ { 3 } \subset \mathbf{R} ^ { 4 }$, the Euler equation becomes $
8 KB (1,192 words) - 16:56, 1 July 2020
Let $D = \{ z \in \mathbf{C} : | z | < 1 \}$ be the unit disc and let, for $1 \leq p < \infty$, $H ^ { p }$ denote the space of holo
...he Poisson kernel (cf. also [[Poisson integral|Poisson integral]]) for the unit disc. A function $\varphi$, defined on $[ - \pi , \pi ]$, belongs to $\oper
18 KB (2,656 words) - 05:17, 15 February 2024
a) $n = 1$. If $M = \Gamma$ is a smooth arc connecting two points of the unit circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and lying inside $\gamma$ an
7 KB (996 words) - 09:31, 3 February 2024
...distributed outside and inside the unit disc without intersecting the unit circle.
with the unit circle is countable, then each bounded uniformly-continuous solution on $ J $
26 KB (3,943 words) - 20:22, 16 January 2024
...\alpha \in S ^ { 1 }$ and all $p \in \bf R$, where $S ^ { 1 }$ is the unit circle in $\mathbf{R} ^ { 2 }$, is known to be:
8 KB (1,283 words) - 20:10, 10 January 2021
For the trigonometric moment problem, $I$ is the unit circle in the complex plane and the polynomials are complex, so that "positive de
6 KB (888 words) - 18:32, 21 December 2020