The inverse operator $K^{-1}$ is constructed from the usual eigenfunction expansion of the operator $K$, but by taking the reciprocal of the eigenvalues in the expansion, rather than the original eigenvalues themselves. Specifically, if $K$ can be represented as:
$$K f = \sum_{n=1}^\infty \lambda_n \langle f, \phi_n \rangle \phi_n,$$
then the inverse $K^{-1}$ is given by:
$$K^{-1} f = \sum_{n=1}^\infty \frac{1}{\lambda_n} \langle f, \phi_n \rangle \phi_n.$$
Verification Steps
To verify that $K^{-1}$ correctly serves as the inverse of $K$, follow these steps:
Compute the Action of the Inverse on the Eigenfunctions:
Calculate $K^{-1} \phi_n$ using the proposed inverse operator. You should obtain:
$$K^{-1} \phi_n = \frac{1}{\lambda_n} \phi_n.$$
Then apply $K$ to this result:
$$K (K^{-1} \phi_n) = K \left(\frac{1}{\lambda_n} \phi_n\right) = \phi_n.$$
Check the Identity Operation:
Verify that applying $K$ to $K^{-1} \phi_n$ yields $\phi_n$ again, which confirms that $K^{-1}$ is the correct inverse with respect to the eigenfunctions.
Full Operator Verification:
Extend this verification beyond just the eigenfunctions to any arbitrary function $f$ in your space. Confirm that:
Numerical Verification (if applicable):
If practical, implement a numerical check using discretized versions of your operator and its inverse. Apply $K$ and $K^{-1}$ to a variety of test functions and verify that the identity holds within a satisfactory numerical tolerance.
crowlogic
commented
2 months ago
Verification of the Inverse Operator
Construction of the Inverse Operator
The inverse operator $K^{-1}$ is constructed from the usual eigenfunction expansion of the operator $K$, but by taking the reciprocal of the eigenvalues in the expansion, rather than the original eigenvalues themselves. Specifically, if $K$ can be represented as:
$$K f = \sum_{n=1}^\infty \lambda_n \langle f, \phi_n \rangle \phi_n,$$
then the inverse $K^{-1}$ is given by:
$$K^{-1} f = \sum_{n=1}^\infty \frac{1}{\lambda_n} \langle f, \phi_n \rangle \phi_n.$$
Verification Steps
To verify that $K^{-1}$ correctly serves as the inverse of $K$, follow these steps:
$$K^{-1} \phi_n = \frac{1}{\lambda_n} \phi_n.$$
Then apply $K$ to this result:
$$K (K^{-1} \phi_n) = K \left(\frac{1}{\lambda_n} \phi_n\right) = \phi_n.$$
Check the Identity Operation: Verify that applying $K$ to $K^{-1} \phi_n$ yields $\phi_n$ again, which confirms that $K^{-1}$ is the correct inverse with respect to the eigenfunctions.
Full Operator Verification: Extend this verification beyond just the eigenfunctions to any arbitrary function $f$ in your space. Confirm that:
$$K(K^{-1} f) = f \quad \text{and} \quad K^{-1}(K f) = f.$$