Pseudo-differential Operator - Definition of Pseudo-differential Operators

Definition of Pseudo-differential Operators

Here we view pseudo-differential operators as a generalization of differential operators. We extend formula (1) as follows. A pseudo-differential operator P(x,D) on Rn is an operator whose value on the function u(x) is the function of x:

\quad P(x,D) u (x) = \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} e^{i (x - y) \xi} P(x,\xi) u(y) \, dy \, d\xi

(2)

where the symbol P(x,ξ) in the integrand belongs to a certain symbol class. For instance, if P(x,ξ) is an infinitely differentiable function on Rn × Rn with the property

for all x,ξ ∈Rn, all multiindices α,β. some constants Cα, β and some real number m, then P belongs to the symbol class of Hörmander. The corresponding operator P(x,D) is called a pseudo-differential operator of order m and belongs to the class

Read more about this topic:  Pseudo-differential Operator

Famous quotes containing the words definition of and/or definition:

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)