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...ated results in the [[mathematical]] study of the regularity of [[harmonic functions]] on sufficiently smooth domains by [[Oliver Dimon Kellogg]]. ...if the boundary data is of class <math>C^{k,\alpha}</math>, then so is the harmonic function itself. ...

...">{{cite web|url=http://ukcatalogue.oup.com/product/9780198503620.do|title=Harmonic Morphisms Between Riemannian Manifolds|work=Oxford University Press}}</ref> ...is a continuous function called the [[dilation (metric space)|dilation]]. Harmonic morphisms are therefore solutions to [[non-linear]] [[over-determined syste ...

...the '''three spheres inequality''' bounds the <math>L^2</math> norm of a [[harmonic function]] on a given [[sphere]] in terms of the <math>L^2</math> norm of t Let <math>u</math> be an harmonic function on <math>\mathbb R^n</math>. Then for all <math>0 < r_1 < r <r_2</ ...

In [[mathematics|mathematical]] [[abstract harmonic analysis|harmonic analysis]], '''Harish-Chandra's ''Ξ'' function''' is a [[special function|s [[Category:Harmonic analysis]] ...

...ny [[holomorphic function]] on the unit disc with positive real part. Such functions had already been characterized in 1907 by [[Constantin Carathéodory]] in te ==Herglotz-Riesz representation theorem for harmonic functions== ...

...ecial functions and [[complex analysis]] being a pioneer in the field of [[harmonic analysis]]. Schuster is known in the field of harmonic analysis for the function that receives his name, whose analytic structure ...

...t harmonic analysis]], '''Harish-Chandra's Schwartz space''' is a space of functions on a [[semisimple Lie group]] whose derivatives are rapidly decreasing, stu ...sely, if ''G'' is connected then the Schwartz space consists of all smooth functions ''f'' on ''G'' such that ...

On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented ...able of the function. This can be obtained by rotation of the basic zonal harmonic <math>Z^{(\ell)}(\theta,\phi).</math> ...

...ion of harmonic functions by harmonic polynomials and by harmonic rational functions|journal=Bull. Amer. Math. Soc.|year=1929|volume=35|issue=2|pages=499–544|do ...ce|approximated uniformly]] on <math>\partial{K}</math> by (real-valued) [[harmonic polynomial]]s in the real variables {{math|x}} and {{math|y}}.<ref>{{cite b ...

With the [[Spherical harmonic#Addition theorem|spherical-harmonic addition theorem]] the equation can be rewritten as *{{Citation|title=Digital Library of Mathematical Functions, Equation 10.60.7 |publisher=National Institute of Standards and Technology ...

...=https://archive.org/details/airyfunctionsapp0000vall/mode/1up |title=Airy functions and applications to physics |date=2004 |publisher=Hackensack, NJ : World Sc |[[Harmonic oscillator|Damped harmonic oscillator]] ...

In [[mathematics]], in the field of [[harmonic analysis]], an '''oscillatory integral operator''' is an [[integral operato ...udied by [[Elias M. Stein|Elias Stein]] and his school.<ref>Elias Stein, ''Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals''. ...

...o a series consisting of radial and spherical parts: the [[solid spherical harmonic]] expansion. ...hogonal group]] in {{mvar|n}} dimensions. This characterization of radial functions makes it possible also to define radial [[distribution (mathematics)|distri ...

...nction in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is ==Statement for harmonic functions== ...

...onic mean]] can all be interpreted as Fréchet means for different distance functions. ===Harmonic mean=== ...

...al|first1=Wladimir Igorewitsch|last1=Mazaew |title=On the growth of entire functions that admit a certain estimate from below |journal=Soviet Math. Dokl. |volum ...armonic Functions Admitting Certain Lower Bounds |encyclopedia=Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications |v ...

...ref>{{Cite journal |last=Wettstein |first=Jerome D. |date=2023 |title=Half-harmonic gradient flow: aspects of a non-local geometric PDE |url=http://www.aimspre * <math>C_0(\mathbb{R}^n)</math> denotes the space of continuous functions <math>f : \mathbb{R}^n \to \mathbb{R}</math> that vanish at infinity, i.e., ...

{{Short description|Family of functions}} ...{{cite book |last1=Pereyra |first1=María |last2=Ward |first2=Lesley |title=Harmonic Analysis: From Fourier to Wavelets |date=2012 |publisher=American Mathemati ...

...ions to the infinity Laplace equation have been identified with the payoff functions from '''randomized tug-of-war''' [[Differential game|games]]. The game theo ...nite or infinite [[graph theory|graph]] <math>G(V,E)</math> is '''discrete harmonic''' on a subset <math>U\subseteq V</math> if ...

...ient condition for a function to be of bounded type (defined in terms of a harmonic majorant), and if <math>\Omega</math> is [[simply connected]] the condition Functions of bounded type are not necessarily bounded, nor do they have a property ca ...