Fourier inversion theorems for integral transforms involving Bessel functions

Technical Summary: Fourier Inversion Theorems for Integral Transforms Involving Bessel Functions

Problem Statement
The paper addresses the derivation and analysis of inversion theorems for generalized Weber-Orr transforms, which are integral transforms involving Bessel functions. While Fourier transforms are typically based on the spectral decomposition of differential operators with trivial kernels, the author notes that operators with non-trivial kernels (specifically those possessing eigenfunctions corresponding to a zero eigenvalue) disrupt standard Fourier properties. In such cases, the standard inversion formula and the Plancherel theorem fail unless the system of generalized eigenfunctions is completed by adding ordinary eigenfunctions associated with the zero eigenvalue.

Historically, inversion theorems for Weber-Orr transforms (first published by Weber in 1873 and later by Titchmarsh in 1923) relied on assumptions of bounded variation for the original function $f$. This paper aims to extend these results to the class of integrable functions ($f \in L^1 \cap L^2$), which is more typical for Fourier analysis, and to rigorously handle cases where the transforms possess non-trivial kernels.

Methodology
The author employs a reverse interaction between functional analysis and partial differential equations (PDEs). Rather than treating the integral transform as the primary object, the paper derives the properties of the Weber-Orr transforms by analyzing the initial-boundary value problems for the heat equation in the exterior of a circle ($r > r_0$).

The methodology proceeds as follows:

1. Operator Association: The transforms are associated with the operator $\Delta_k = \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial}{\partial r}) - \frac{k^2}{r^2}$.
2. Laplace Transform: The heat equation $\partial_t w - \Delta_k w = 0$ is transformed into a modified Bessel equation (Poisson equation with a parameter) using the Laplace transform.
3. Resolvent Analysis: The solution is constructed using the resolvent $R(\tau) = (\Delta_k - \tau I)^{-1}$. The author analyzes the branch cut of the resolvent along the negative real axis (continuous spectrum) and investigates the behavior at $\tau = 0$.
4. Contour Integration: The inverse Laplace transform is evaluated by deforming the integration contour to a Hankel-type contour around the branch cut. This process isolates the continuous spectrum contribution (leading to the integral transform) and, crucially, the residue at $\tau = 0$ when the kernel is non-trivial.
5. Asymptotic Analysis: Asymptotic forms of Bessel functions ($J_k, Y_k, I_k, K_k$) are used to establish bounds and derive the explicit jump discontinuities across the branch cut, which yield the kernel of the integral transform.

Key Contributions and Results

1. Classical Weber-Orr Transform ($W_{k,k}$):

   • The paper proves an inversion theorem for $W_{k,k}$ under the condition $f\sqrt{r} \in L^1(r_0, \infty) \cap L^2(r_0, \infty)$.
   • It establishes that for this transform, the kernel is trivial, and the standard inversion formula holds almost everywhere:
     $$f(r) = W^{-1}_{k,k}[W_{k,k}[f]](r)$$
   • A Plancherel identity is proven: $\|f\|_{L^2} = \|W_{k,k}[f]\|_{L^2}$.

2. Generalized Weber-Orr Transform ($W_{k,k\pm1}$):

   • The paper investigates transforms with indices $k, k\pm1$, which correspond to Robin-type boundary conditions.
   • Non-Trivial Kernel: It is demonstrated that for specific ranges of $k$ (specifically $k > 1$ for $W_{k,k-1}$ and $k < -1$ for $W_{k,k+1}$), the operator $\Delta_k$ possesses a non-trivial kernel containing eigenfunctions of the form $r^{-k}$ or $r^k$.
   • Modified Inversion Formula: The presence of the kernel necessitates an additional term in the inversion formula. For $k > 1$, the inversion is given by:
     $$f(r) = W^{-1}_{k,k-1}[W_{k,k-1}[f]](r) + 2(k-1)r_0^{2(k-1)}r^{-k} \int_{r_0}^\infty s^{-k+1}f(s)ds$$
   • Plancherel-Parseval Identity: The standard Plancherel equality is replaced by an identity that includes the sum of squares of the Fourier coefficients corresponding to the zero eigenvalue:
     $$\|f\|^2 = \|W[f]\|^2 + \sum (f, e_k)^2$$
     This confirms the necessity of adding the projection onto the kernel to preserve energy conservation.

3. Bessel-type Inequality:

   • A corollary establishes a Bessel-type inequality for the generalized transforms, showing that the norm of the transformed function is bounded by the norm of the original function.

Significance and Claims
The paper claims to provide a "new approach" to the study of classical and generalized Weber-Orr transforms. Its primary significance lies in:

• Completeness of Derivation: It offers a complete derivation of inversion formulas within the class of integrable functions ($L^1 \cap L^2$), replacing the classical pointwise equality (valid for bounded variation) with an identity holding almost everywhere.
• Handling Non-Trivial Kernels: It explicitly addresses the spectral decomposition of operators with non-trivial kernels, demonstrating how the continuous spectrum must be supplemented by eigenfunctions corresponding to the zero eigenvalue to satisfy Plancherel-Parseval type identities.
• Hydrodynamic Application: The author notes that these transforms (specifically $W_{k,k-1}$) have already been applied to solve non-stationary Stokes and Navier-Stokes problems in exterior domains, effectively acting as a "hydrodynamic Fourier transform." The results in this paper provide the rigorous theoretical foundation (invertibility and energy conservation) required for these applications.

The paper does not propose new experimental applications or future implications beyond the mathematical framework established, focusing instead on the rigorous proof of invertibility and spectral properties for these specific integral transforms.