defined on the basic spaceFourier transform.For any function, define its Fourier transform as
agreed mark
Fourier transform has the following properties:
(1) Fourier transform is a linear transformation.
(2) Fourier transformation changes the differential operation into an operation multiplied by a power function, that is
(3) Fourier transformation changes the operation of multiplying the power function into a differential operation, that is
(4) Fourier transform willConvolution operationbecomeMultiplication, otherwise it willMultiplicationbecomeConvolution operation.i.e.
Theorem 1 The Fourier transformation establishes an isomorphic correspondence to .
proveFirst prove that for any one, it still belongs to after Fourier transformation. In fact, for any heavy index, the integral equation
Just have it
Since, it is bounded for any, and thus. From the same equation, it can be seen that when, there is also, so the Fourier transformation is an isomorphic correspondence to. It can also be proved that the inverse Fourier transformation is also an isomorphic correspondence to (one-to-one, and the linear structure and topological structure remain unchanged).
Theorem 2 Regarding the Fourier transformation, the Parseval equation is established:
Example 1 The Fourier transform of the function is.
Definition 1 For any generalized function, define its Fourier transformation as
Theorem 3 The Fourier transformation establishes an isomorphic correspondence to .
Example 2 Find the Fourier transformation of a function. For any, it is true
Example 3 Find the Fourier transform of a function. For any, note, then, and we have
Example 4 Find the Fourier transform.
Note that for any , it is still recorded as, then
Example 5 Find the Fourier transform of a generalized function.
Note that for any , the integral is meaningful, so let
Therefore, that is.
Theorem 4 If, then.
Theorem 5 Paley-Wiener theorem.
(1) If, the support is in, then the Fourier-Laplace transformation satisfies the condition (P1): there exists such that
And if the analytic function satisfies condition (P1), it must be a certain Fourier-Laplace transformation, and the support of and is in .
(2) If, the support is in, then the Fourier-Laplace transformation satisfies the condition (P2): for any, exists, so that
And if the analytic function satisfies condition (P2), it must be a certain Fourier-Laplace transformation, and the support of and is in .
Suppose a linear differential operator with coefficients is given in space, expressed as
Then it can be known from the properties of Fourier transform
Definition 2 Assume a function, and for any heavy index, it holds
Where is a constant, it is called a class function, denoted as .
Definition 3 If function, then the linear continuous mapping that can be defined is a symbol of quasi-differential operator
When it depends only on variables, the operator is also called a Fourier multiplier.
Theorem 6The quasi-differential operator is a linear continuous operator of . It can also be uniquely extended to become a linear continuous operator of , where , the linear continuous operator of , where , the convergence of in sequenceweak convergenceunderstanding of meaning.
The function class is a generalization of polynomials. Correspondingly, the quasi-differential operator class is broader than the differential operator class, and its entirety constitutes an operator algebra. An important fact is that the operations of quasi-differential operators can often be realized through their symbolic operations. This overcomes the limitation that differential operators can only perform addition and multiplication (composition). The solution of partial differential equations is regarded as the inversion of partial differential operators, and is realized through symbolic operations.
Machine-translated from the Chinese original 现代偏微分方程笔记(Fourier变换) (published 2024-01-14). WeChat original ↗