Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory.
In simpler terms, the definition of a pseudo-differential operator depends on the Fourier Transform. This topic is actually covered long after the introduction of the Fourier Transform. The Fourier Transform is covered at varying levels of rigor beginning with a cursory introduction in a standard second year differential equations course focusing on applications and progressing to a rigorous introduction often found in advanced undergraduate to beginning graduate courses in Fourier Analysis which relies on the use of mathematical analysis rather than exclusively univariate or multivariate calculus.
The study of pseudo-differential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza.
They played an influential role in the first proof of the Atiyah–Singer index theorem. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of Pseudo-differential operators.
Consider a linear differential operator with constant coefficients,
which acts on smooth functions with compact support in Rn. This operator can be written as a composition of a Fourier transform, a simple multiplication by the polynomial function (called the symbol)
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