Discrete Bessel and Mathieu functions

This contribution introduces discrete Bessel and Mathieu functions by applying the discrete dihedral group to the Helmholtz equation in polar and elliptic coordinates, and demonstrates that these functions serve as high-precision approximants of finite sums of their continuous counterparts, while preserving essential relations of special functions.

The Gist

Imagine you are trying to describe a complex wave, such as a wave spreading across a pond or a sound wave moving through the air. In the world of physics, mathematicians use special tools called "functions" to precisely capture how these waves behave. Two of the most well-known tools for this purpose are the Bessel functions (used for circular waves) and the Mathieu functions (used for oval or elliptical waves).

Think of these continuous functions as a smooth, unbroken line drawn on a sheet of paper. They are perfect, flowing, and exist at every single point along the curve. Computers, however, do not work with smooth lines; they work with points. They can only process a finite number of points.

This article is about creating a new set of mathematical tools that are the "point version" of these smooth lines. The authors, Kenan Uriostegui and Kurt Bernardo Wolf, have figured out how to replace the smooth, infinite world of these waves with a finite, digital world made of discrete points, while preserving the essential magic of the original waves.

Here is how they did it, broken down into simple concepts:

1. The Circle versus the Polygon

In the real world, a circle is continuous. One can rotate around it at any arbitrary angle. However, imagine standing on a clock face with only 12 numbers. You can stand only at 12 specific positions.

The authors adapted the standard method for describing waves (which involves rotating around a full circle) so that the infinite number of possible angles is replaced by a fixed number of steps, say $N$ steps.

• The old way: One integrates (sums) the wave over every possible angle from 0 to 360 degrees.
• The new way: One considers only $N$ specific, evenly distributed angles (like the hours on a clock) and sums the values at exactly those positions.

They call these new tools Discrete Bessel functions. They behave exactly like the famous smooth Bessel functions but are constructed from a finite list of numbers rather than a smooth curve.

2. The Challenge of Ovals (Ellipses)

The article goes one step further. While circles are simple, what about ovals (ellipses)? Waves in oval-shaped spaces or around oval objects are described by Mathieu functions.

The authors applied the same "point" logic to these oval waves. They took the smooth oval coordinate system and placed a grid of discrete points along the boundary of the oval.

• They created Discrete Mathieu functions that exist at these specific points.
• Just as with the circles, they found that these "point-based" functions mimic the "smooth" originals incredibly well.

3. The "Magic" of Approximation

The most exciting part of their discovery is how close these "point" versions come to the "smooth" originals.

• The Analogy: Imagine taking a high-resolution photograph of a smooth painting. If you zoom in far enough, you see pixels. However, if you step back, the pixels merge into an image that looks exactly like the smooth painting.
• The Result: The authors found that their discrete functions, for a certain range of values, come so close to the continuous originals that the difference is practically invisible (smaller than one part in a quadrillion).

They proved that a wave moving in a specific direction can be described by a finite sum of these discrete functions, and it will look almost identical to the real wave.

4. Why This Matters (According to the Article)

The authors emphasize that this is not just about making mathematics simpler; it is about changing the fundamental symmetry of the problem.

• Continuous Symmetry: In the real world, one can rotate an object by a tiny amount, and the laws of physics remain the same.
• Discrete Symmetry: In their new model, one can rotate the object only by specific "steps" (like turning a dial to the next notch).

They show that the mathematics works wonderfully even with this "step-by-step" restriction. The "Discrete Bessel functions" and "Discrete Mathieu functions" preserve the key relationships and rules that the smooth versions possess.

Summary

In short, the authors have translated the complex, smooth mathematics used to describe waves in circles and ovals into a language that computers love: finite lists of numbers.

They have built a bridge between the infinite, smooth world of analysis and the finite, pixelated world of digital computation. Their "Discrete Bessel functions" and "Discrete Mathieu functions" are the digital twins of the classic mathematical giants, precise enough to serve as perfect substitutes in many scenarios, all while respecting the underlying geometry of the universe.

Technical Summary

1. Problem Statement

The paper addresses the need for discrete analogs of continuous special functions used to solve the two-dimensional Helmholtz equation.

• Context: The Helmholtz equation, $(\nabla^2 + \kappa^2)f = 0$, is separable in four coordinate systems (Cartesian, polar, parabolic, elliptic) due to the symmetries of the Euclidean group (translations and continuous rotations/reflections).
• Limitation: While continuous Bessel and Mathieu functions are the standard for polar and elliptic coordinates respectively, there is a need for discrete versions that either approximate these functions for computational efficiency or serve to solve difference equations arising in discrete physical systems (e.g., digital signal processing, discrete wave propagation).
• Gap: Previous attempts to define discrete Bessel functions existed but were incomplete or lacked a unified group-theoretic framework. Furthermore, discrete Mathieu functions had not been systematically defined using a similar symmetry-breaking approach.

2. Methodology

The authors employ a symmetry-based discretization strategy, replacing the continuous rotational symmetry of the Euclidean group with a discrete dihedral group ($D_N$), consisting of $N$ discrete rotations and reflections.

• Discretization of the Angular Variable:

  • The continuous circle $S^1$ (angle $\phi \in (-\pi, \pi]$) is replaced by a set of $N$ equidistant points $S^1_{(N)}$, defined by $\phi_m = 2\pi m/N$ for $m \in \{0, 1, \dots, N-1\}$.
  • Continuous Fourier integrals are replaced by finite $N$-point Fourier sums.
  • The scalar product of functions is changed from an integral over the circle to a discrete sum over the $N$ points.

• Application to Polar Coordinates (Bessel Functions):

  • The authors revise the definition of discrete Bessel functions $B^{(N)}_n(\rho)$ by expanding plane waves into a finite sum of polar components.
  • They utilize the orthogonality of discrete phase functions $\Phi^{(N)}_n(\phi_m) = N^{-1/2} \exp(in\phi_m)$ to isolate the radial component.

• Application to Elliptic Coordinates (Mathieu Functions):

  • Elliptic coordinates $(\varrho, \psi)$ are introduced, where $\psi$ is the angular variable and $\varrho$ is the radial variable.
  • The angular variable $\psi$ is discretized onto $N$ points, while the radial variable $\varrho$ remains continuous (as it is not cyclic).
  • Discrete Angular Mathieu Functions: Defined by truncating the infinite Fourier series of continuous Mathieu functions ($ce_n, se_n$) to a finite sum of $N$ terms, with coefficients determined by discrete Fourier transforms.
  • Discrete Radial Mathieu Functions: Derived by expanding the Helmholtz solution in the discrete angular basis and extracting the radial factor using the orthogonality of the discrete angular functions.

3. Main Contributions

• Unified Framework: The paper establishes a rigorous method for generating discrete special functions by replacing the continuous rotation subgroup of the symmetry group with a finite cyclic subgroup.
• Definition of Discrete Mathieu Functions: The authors provide the first systematic definition of discrete Mathieu functions of the first kind (angular) and second kind (radial) based on the $N$-point discrete Fourier transform.
• Analytical Relations: They derive exact discrete analogs of known continuous relations, including:
  • Linear relations between even and odd discrete Bessel functions (analogous to the Graf addition theorem).
  • Orthogonality relations for discrete Mathieu functions under the discrete scalar product.
  • Parity and cyclicity properties ($B^{(N)}_n = B^{(N)}_{n \pm N}$).
• Coefficient Mapping: They establish the correspondence between the coefficients of the discrete Fourier series ($a_n, b_n$) and the continuous Fourier coefficients ($A_n, B_n$), showing that $a_n \approx A_n/2$ (for $n \neq 0$) holds.

4. Results

• High-Precision Approximation:
  • Numerical verifications show that the discrete Bessel functions $B^{(N)}_n(\rho)$ approximate the continuous Bessel functions $J_n(\rho)$ with an error on the order of $10^{-16}$ in the range $0 \le n + \rho < N$.
  • Similarly, the discrete angular Mathieu functions $ce^{(N)}_n(\psi_m, q)$ at the discrete points $\psi_m$ agree with their continuous counterparts $ce_n(\psi, q)$ with errors $< 10^{-16}$.
• Radial Behavior:
  • Discrete radial Mathieu functions $Ce^{(N)}_n(\varrho, q)$ and $Se^{(N)}_n(\varrho, q)$ agree closely with continuous radial Mathieu functions for $\varrho \in [0, \pi)$.
  • Oscillation Phenomenon: Beyond $\varrho \approx \pi$, the discrete radial functions exhibit wild oscillations (Gibbs-like phenomena), indicating the limits of the approximation when the argument exceeds the sampling range.
• Specific Cases:
  • The paper successfully derives explicit summation formulas for the radial functions for even and odd parities.
  • For the specific case of $Se^{(N)}_{2n+2}$, for which a direct integral analog is missing in standard literature, the authors propose a numerical relation $Se^{(N)}_{2n+2} \approx -i se_{2n+2}(i\varrho, q)$, which holds with high accuracy ($< 10^{-14}$) for small $\varrho$.

5. Significance

• Computational Efficiency: These discrete functions offer a way to solve wave propagation problems in systems with discrete symmetries (e.g., photonic crystals, digital antenna arrays, or resonant microcavities) without resorting to full numerical PDE solvers.
• Theoretical Insight: The work bridges the gap between continuous harmonic analysis and discrete signal processing, demonstrating that the "discretization" of the symmetry group naturally leads to valid approximants of special functions.
• Future Applications: The authors suggest that these functions can describe wave fields in 2D microcavities fed by a finite number of activation channels. The methodology is extendable to 3D, where the rotation group could be reduced to polyhedral subgroups, potentially describing wave fields with specific directional constraints.

In summary, the paper successfully constructs a "discrete calculus" theory for Bessel and Mathieu functions by exploiting finite group theory, provides high-precision approximations for specific ranges, and lays a foundation for solving discrete Helmholtz problems in both polar and elliptic geometries.