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Introduction

  • Fabio Nicola
  • Luigi Rodino
Part of the Pseudo-Differential Operators book series (PDO, volume 4)

Abstract

To give an introduction to the contents of the book, let us consider initially the basic models to which our pseudo-differential calculus will apply, namely the linear partial differential operators with polynomial coefficients in ℝ d
$$ P = \sum {c_{\alpha \beta } x^\beta D^\alpha } , x \in \mathbb{R}^d , c_{\alpha \beta } \in \mathbb{C}, $$
(I.1)
where in the sum (α,β) ∈ ℕ d × ℕ d run over a finite subset of indices. A natural setting for P is given by the Schwartz space \( S\left( {\mathbb{R}^d } \right) \) and its dual \( S'\left( {\mathbb{R}^d } \right) \). These spaces are invariant under the action of the Fourier transform
$$ Fu\left( \xi \right) = \hat u\left( \xi \right) = \int {e^{ - ix\xi } u\left( x \right)d--x, } with d--x = \left( {2\pi } \right)^{ - \frac{d} {2}} dx. $$
(I.2)
Note also that the conjugation \( FPF^{ - 1} \) gives still an operator of the form (I.1).

Keywords

Harmonic Oscillator Principal Symbol Schwartz Space Fredholm Property Holomorphic Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Fabio Nicola
    • 1
  • Luigi Rodino
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly
  2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

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