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Pseudodifferential Operator

A pseudodifferential operator is an operator represented locally by a Fourier integral

Pu(x)=1/((2pi)^n)int_(R^n)e^(ix·xi)p(x,xi)u^^(xi)dxi,

where the symbol p(x,xi) satisfies prescribed estimates in x and xi. A differential operator is the special case in which the symbol is a polynomial in xi. Allowing more general symbols provides parametrices for elliptic differential operators and supports the study of propagation of singularities.

See also

Differential Operator, Elliptic Partial Differential Equation, Fourier Transform

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References

Hörmander, L. The Analysis of Linear Partial Differential Operators III. New York: Springer-Verlag, 1985.

Cite this as:

Weisstein, Eric W. "Pseudodifferential Operator." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PseudodifferentialOperator.html

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