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Definition 2024

differential_operator

differential operator

English

Noun

differential operator ‎(plural differential operators)

(mathematics, analysis) Differentiation, regarded as a mapping of a function to its derivative.
- 2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328,
  […] these conditions appeared in the very first papers on ordinary differential operators.
- 2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27,
  
  A differential operator $P\left(D\right)$ is hypoelliptic if for any domain ⊂ $\mathbb {R} ^{n}$ any solution $u$ of the equation $P\left(D\right)u=0$ from the class ${\mathfrak {D}}'\left(\Omega \right)$ is a function from $C^{\infty }$ (ω) for any open set ω ⊂⊂ $\Omega$ .
  
  A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].
- 1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251,
  Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.

Related terms

partial differential operator
pseudodifferential operator