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- #redirect [[subharmonic function]] ...34 bytes (3 words) - 18:39, 25 November 2005
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- {{Short description|Topology in the study of subharmonic functions}} ...n]] topology, but with the advent of upper [[semi-continuous]] subharmonic functions introduced by [[F. Riesz]], the fine topology became the more natural tool ...4 KB (684 words) - 12:02, 23 October 2022
- {{Short description|Class of mathematical functions}}{{Moreinline|date=June 2025}} ...perharmonic''' functions are important classes of [[function (mathematics)|functions]] used extensively in [[partial differential equations]], [[complex analysi ...12 KB (1,831 words) - 16:56, 17 June 2025
- ...h> (where <math>n\ge 2</math>) is a polar set if there is a non-constant [[subharmonic function]] ...equivalent) ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and <math>-\infty</math> by <math>\infty</math> in the ...2 KB (332 words) - 07:33, 26 February 2024
- ...unctions (which are defined on a [[Riemannian manifold]]) plurisubharmonic functions can be defined in full generality on [[complex analytic space]]s. the function <math>z \mapsto f(a + bz)</math> is a [[subharmonic function]] on the set ...8 KB (1,243 words) - 12:27, 19 December 2024
- ...rmonic function|subharmonic]] and [[Superharmonic function|superharmonic]] functions. ...*</sup> with respect to the sphere {{nowrap|''S''(0, ''R'')}} is harmonic, subharmonic or superharmonic in ''D''<sup>*</sup>. ...3 KB (516 words) - 00:20, 3 December 2023
- | known_for = Works on analytical functions, [[Luzin-Privalov theorems]]. ...blems jointly with [[Nikolai Luzin|Luzin]]. In 1934 he studied subharmonic functions, building on the work of [[Marcel Riesz|Riesz]]. ...3 KB (405 words) - 11:28, 6 November 2024
- ...ions with a rich structure are, in addition to the convex functions, the [[subharmonic function]]s and the [[plurisubharmonic function]]s. Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also o ...2 KB (275 words) - 21:13, 12 May 2024
- ...= [[Radó's theorem (Riemann surfaces)]]<br>[[Radó's theorem (harmonic functions)]]<br>[[Radó–Kneser–Choquet theorem]]<br>[[Covering problem of Rado]]<br>[[ ...and its [[non-computable function|non-computability]] ("On Non-Computable Functions"). ...6 KB (860 words) - 21:53, 18 April 2025
- ...ector calculus]] that acts as the inverse to the negative [[Laplacian]] on functions that are smooth and decay rapidly enough at infinity. As such, it is a fund ...oisson equation in suitably regular domains, and for suitably well-behaved functions ''f'': one first applies a Newtonian potential to obtain a solution, and th ...6 KB (916 words) - 23:10, 6 May 2025
- {{short description|Functions in mathematics}} {{about|harmonic functions in mathematics|harmonic function in music|diatonic functionality}} ...23 KB (3,593 words) - 07:50, 28 August 2025
- {{Short description|Harmonic functions as solutions to Laplace's equation}} ...e time, namely gravity and the electrostatic force, could be modeled using functions called the [[gravitational potential]] and [[electrostatic potential]], bot ...10 KB (1,426 words) - 10:40, 13 March 2025
- In the theory of complex functions, it is known that the [[Absolute value|modulus]] (absolute value) of a [[Ho ...similar fashion to [[subharmonic function|subharmonic]] and superharmonic functions. ...12 KB (1,924 words) - 15:25, 28 June 2025
- ...ined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the [[Lp space|''L<sup>p</sup>'' spaces]].{{sf ...h>H^p</math> for <math>0 < p < \infty</math> is the class of [[holomorphic functions]] <math>f</math> on the open [[unit disk]] <math> \mathbb{D} = \{ z\in \mat ...27 KB (4,509 words) - 06:09, 2 April 2025
- ...tency of the above-mentioned approximation can be shown for highly regular functions, such as <math> u \in C^4(\Omega) </math>. ==== Subharmonic ==== ...21 KB (3,258 words) - 00:59, 20 May 2025
- ...nce des fonctions analytiques et sous-harmoniques ([[complex analysis]], [[subharmonic function]]s) ...7 KB (933 words) - 03:37, 10 November 2024
- ...clude [[Graph of a function|graphs]] of [[Multivalued function|multivalued functions]] such as √''z'' or log(''z''), e.g. the subset of pairs {{nowrap|1=(''z'', === Functions === ...26 KB (3,774 words) - 10:43, 20 March 2025
- ==Submartingales, supermartingales, and relationship to harmonic functions{{anchor|Submartingales and supermartingales}}== ...ory, a [[subharmonic function]] ''f'' satisfies Δ''f'' ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the ...23 KB (3,708 words) - 21:44, 29 May 2025
- ...o State University]]; published: "On the Problem of Plateau", "Subharmonic Functions", in the ''[[Bell System Technical Journal]]'' the [[Busy Beaver]] problem ...8 KB (1,049 words) - 16:14, 26 October 2023
- {{short description|Property of functions which is weaker than continuity}} ...ble. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.<ref>Willard, p. 49, problem 7K</ref> For example the f ...41 KB (6,512 words) - 11:51, 17 September 2025
- The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, when the boundary is su ...ing the [[Perron method]], which relies on the [[maximum principle]] for [[subharmonic function]]s. This approach is described in many text books.<ref>See for exa ...14 KB (2,039 words) - 13:00, 12 June 2025