Symbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.
Let P be a linear differential operator of order k on the Euclidean space Rd. Then P is a polynomial in the derivative D, which in multi-index notation can be written
The total symbol of P is the polynomial p:
The leading symbol, also known as the principal symbol, is the highest degree component of σP :
and is of importance later because it is the only part of the symbol that transforms as a tensor under changes to the coordinate system.
The symbol of P appears naturally in connection with the Fourier transform as follows. Let ƒ be a Schwartz function. Then by the inverse Fourier transform,
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