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- Riesz_potential abstract "In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by</dl>where the constant is given byThis singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α. If p > 1, then the rate of decay of f and that of Iαf are related in the form of an inequality (the Hardy–Littlewood–Sobolev inequality)More generally, the operators Iα are well-defined for complex α such that 0 < Re α < n.The Riesz potential can be defined more generally in a weak sense as the convolutionwhere Kα is the locally integrable function:The Riesz potential can therefore be defined whenever f is a compactly supported distribution. In this connection, the Riesz potential of a positive Borel measure μ with compact support is chiefly of interest in potential theory because Iαμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rn.Consideration of the Fourier transform reveals that the Riesz potential is a Fourier multiplier. In fact, one hasand so, by the convolution theorem,The Riesz potentials satisfy the following semigroup property on, for instance, rapidly decreasing continuous functionsprovidedFurthermore, if 2 < Re α <n, thenOne also has, for this class of functions,".
- Riesz_potential wikiPageID "6779393".
- Riesz_potential wikiPageRevisionID "584241223".
- Riesz_potential first "E.D.".
- Riesz_potential hasPhotoCollection Riesz_potential.
- Riesz_potential id "R/r082270".
- Riesz_potential last "Solomentsev".
- Riesz_potential title "Riesz potential".
- Riesz_potential subject Category:Fractional_calculus.
- Riesz_potential subject Category:Partial_differential_equations.
- Riesz_potential subject Category:Potential_theory.
- Riesz_potential subject Category:Singular_integrals.
- Riesz_potential type Abstraction100002137.
- Riesz_potential type Calculation105802185.
- Riesz_potential type Cognition100023271.
- Riesz_potential type Communication100033020.
- Riesz_potential type DifferentialEquation106670521.
- Riesz_potential type Equation106669864.
- Riesz_potential type HigherCognitiveProcess105770664.
- Riesz_potential type Integral106015505.
- Riesz_potential type MathematicalStatement106732169.
- Riesz_potential type Message106598915.
- Riesz_potential type PartialDifferentialEquation106670866.
- Riesz_potential type PartialDifferentialEquations.
- Riesz_potential type ProblemSolving105796750.
- Riesz_potential type Process105701363.
- Riesz_potential type PsychologicalFeature100023100.
- Riesz_potential type SingularIntegrals.
- Riesz_potential type Statement106722453.
- Riesz_potential type Thinking105770926.
- Riesz_potential comment "In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space.".
- Riesz_potential label "Riesz potential".
- Riesz_potential sameAs リースポテンシャル.
- Riesz_potential sameAs m.0gnhnc.
- Riesz_potential sameAs Q7333166.
- Riesz_potential sameAs Q7333166.
- Riesz_potential sameAs Riesz_potential.
- Riesz_potential wasDerivedFrom Riesz_potential?oldid=584241223.
- Riesz_potential isPrimaryTopicOf Riesz_potential.