Appl. Math. Inf. Sci. 15, No. 3, 307-315 (2021) 307 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/150308 Discrete Bessel and Mathieu Functions Kenan Uriostegui∗ and Kurt Bernardo Wolf Institute of Physical Sciences, National Autonomous University of M´exico, Morelos 62251, M´exico Received: 2 Feb. 2021, Revised: 2 Apr. 2021, Accepted: 12 Apr. 2021 Published online: 1 May 2021 Abstract: The two-dimensional Helmholtz equation separates in elliptic coordinates based on two distinct foci, a limit case of which includes polar coordinate systems when the two foci coalesce. This equation is invariant under the Euclidean group of translations and orthogonal transformations; we replace the latter by the discrete dihedral group of N discrete rotations and reflections. The separation of variables in polar and elliptic coordinates is then used to define discrete Bessel and Mathieu functions, as approximants to the well-known continuous Bessel and Mathieu functions, as N-point Fourier transforms approximate the Fourier transform over the circle, with integrals replaced by finite sums. We find that these ‘discrete’ functions approximate the numerical values of their continuous counterparts very closely and preserve some key special function relations. Keywords: Bessel functions, Mathieu functions, Fourier analysis, Group theory, Numerical approximation and analysis 16 1 Introduction 10− ) the corresponding continuous ones over a region, roughly 0 n + ρ < N. ≤ The role of the Euclidean group of translations, Several authors have introduced functions that reflections and rotations in the determination of the approximate the well-known continuous Bessel functions coordinate systems that separate the solutions of the Jn(ρ) for the purpose of reducing computation time, or to two-dimensional Helmholtz equation is well known from provide new classes of solutions to difference equations the work by Willard Miller Jr. [1]. This symmetry that will share some of their salient properties [4,5,6]. accounts for their separability in four coordinate systems: Our approach follows the well known approximation Cartesian, polar, parabolic and elliptic. Only the elliptic afforded by the N-point finite Fourier transform to the system is generic; when the two foci coalesce, this system integral Fourier transform over the circle. This is done for becomes the polar one with angular and radial polar and elliptic coordinates, and introduces both coordinates; when one focus departs to infinity the system ‘discrete’ Bessel and Mathieu functions. These functions, becomes parabolic; and when both foci do, it becomes we should emphasize, differ from those proposed in the Cartesian. works cited above, which are also distinct in definition The polar decomposition was used by Biagetti et al. and purpose among themselves. By construction, it will [2] to first introduce a discrete version of Bessel functions follow that under N ∞, these discrete functions become based on an expansion of plane waves into a finite number the continuous ones,→ although this limit requires further of polar components —that was not quite complete. This mathematical precision, as it may involve Gibbs-type was properly completed in Ref. [3], defining discrete N oscillation phenomena that we cannot address here. Bessel functions Bn (ρ), which approximate the usual continuous Bessel functions Jn(ρ) by replacing Fourier In Sect. 2, we present this discretization method and a series over a circle S 1 by the finite Fourier transform on resum´eof the results in Ref. [3] for Bessel functions, to N equidistant points on that circle, note that the discrete functions thus defined approximate the continuous ones remarkably well. In Sect. 3, we apply S 1 S 1 S 1 θm = 2πm/N, m 0,1,...,N 1 =: (N), (1) the strategy of replacing harmonic analysis on by (N) ∈{ − } to define discrete approximants to the Mathieu functions where m is counted modulo N. It was found that these of first and second kind in the elliptic coordinate system. discrete functions approximated very closely (of the order All relations are backed by numerical verification. In the ∗ Corresponding author e-mail: [email protected] c 2021 NSP Natural Sciences Publishing Cor. 308 K. Uriostegui, K. B. Wolf: Discrete Bessel and Mathieu functions concluding Sect. 4, we provide some further connections of a continuous function, or as the index for the list of and preliminary conclusions. components of an N-cyclic vector. In either case, the inner (1) (2) product of two discrete functions fn and fn is naturally N 1 2 Continuous and discrete Bessel functions (1) (2) − (1) (2) ( f , f )(N) := ∑ fn ∗ fn , (8) The Helmholtz equation for wavefields f (x,y) of (fixed) n=0 real wavenumber κ R is ∈ and it is clear that the N ∞ limit will lead back from the discrete to the continuum,→ with the approximationsand (∂ 2 + ∂ 2 + κ2) f (x,y)= 0, (2) x y limits familiar from Fourier theory. The Helmholtz equation (2) in polar coordinates, with ∂ ∂ ∂ and x y R2. In this section we follow z / z ( , ) multiplied by r2, the well≡ known case of polar∈ coordinates, 2 2 2 2 x = r cosθ, y = r sinθ, (3) (r ∂r + r∂r + ∂φ + κ ) f (r,φ)= 0, (9) r [0,∞), θ ( π,π]= S 1. ∈ ∈ − shows that solutions can be factored into a function of the A key assumption is a Hilbert space structure for the radius times a function of the angle as solutions f (x,y) by which one can write them as the f (r,φ) = R(r)Φ(φ), while (5) implies that solutions two-dimensional Fourier transform, Φ(φ) for the angular factor will determine a corresponding radial factor R(r). An orthonormal and 1 2 f (x,y)= dκx dκy expi(xκx +yκy) f (κx,κy). (4) complete set of eigenfunctions of ∂φ over the circle 2π ZZR2 S 1 1/2 e φ is the set of phases Φn(φ) := (2π)− exp(inφ), The Helmholtz equation (2) is then correspondingly with∈ integer n 0, 1,... , and inner products transformed to a conjugate space κ κ R2 where it (Φ ,Φ ) = δ .∈ When { ± the domain} of these functions is ( x, y) n n′ n,n′ 2 2 2 ∈ ◦ S 1 S 1 reads (κ κ κ ) f (κx,κy) = 0, which we can also restricted from φ to φm N as in (1), we retain − x − y ∈ ∈ ( ) refer to polar coordinates κx = κ cosφ, κy = κ sinφ, with the subset of N functions on the N points in S 1 , given e (N) the surface element dκx dκy = κ dκ dφ. The solutions to by the Fourier-transformed Helmholtz equation are thus 1 1 2πimn reduced by a Dirac δ-distributions in the radius [1], as Φ (N) φ : exp inφ exp (10) 1 n ( m) = ( m)= f (κx,κy) = √2πκ− δ(κ κ) f (φ), with f (φ) a √N √N N − ◦ ◦ function on the φ-circle S 1 of radius κ, that we write (N) e = Φn N (φm), again κ, understanding that ite is the fixed wavenumber. ± The Helmholtz solutions (4) thuse acquire the labeled by the cyclic subset n 0,1,...,N 1 , that are single-integral form also orthonormal under the common∈{ inner product− } (8) for S 1 discrete functions on (N), and complete: 1 N 1 f (x,y)= dφ expiκ(xcosφ + ysinφ) f (φ), N − S 1 (N) ( ) (N) (N) √2π Z ◦ (Φn ,Φ )(N) = δn,n , Φn (φm)∗ Φn (φm )= δm,m . n′ ′ ∑ ′ ′ (5) n=0 with the Hilbert space structure based on the inner product (11) of functions f (1)(φ) and f (2)(φ) on the circle, Returning to (5) with (x,y) in the polar coordinates (r,θ) ◦ ◦ of (4), and taking for f (φm) the basis functions (11) on S◦ 1 (1) (2) (1) (2) the discrete points of (N), we write the N solutions to ( f , f ) := dφ f (φ)∗ f (φ). (6) ◦ ◦ ◦ ZS 1 ◦ ◦ the discretized Helmholtz equation, labeled by cyclical n 0,1,...,N 1 , as ∈ It is here that we reduce the continuous circle Fourier { − } transform to the N-point discrete Fourier transform, from fn(r,θk) S 1 to S 1 , replacing integrals by summations and the (N) = 1 exp[iκr(cosθ cosφ + sinθ sinφ )]Φ (N)(φ ) S 1 S 1 √N ∑ k m k m n m continuous variable φ with φm (N), as m S 1 ∈ (N) ∈ ∈ 1 = N ∑ exp[iκr cos(θk φm)]exp(inφm) 2π N, m S 1 − dφ F (φ) F(φm), ↔ (7) ∈ (N) 1 ∑ ZS ◦ ↔ φm = 2πm/N, in θ π 2 m S 1 e ( k+ / ) ∈ (N) = N ∑ exp(iκr sinϕm)exp( inϕm), m S 1 − ∈ (N) for m 0,1,...,N 1 counted modulo N; the set of N (12) ∈{ − } 1 discrete angles φm are thus equidistant by 2π/N. The having replaced ϕm := θk φm + π in the summation over − 2 functions f (φm) fm can be interpreted as sample points the N discrete points on the circle. ≡ c 2021 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 15, No. 3, 307-315 (2021) / www.naturalspublishing.com/Journals.asp 309 (N) Fig. 1: The ‘discrete’ Bessel functions Bn (ρ) on continuous intervals 0 ρ (2N 1) (gray lines), vs. the ‘continuous’ Bessel ≤ ≤ − functions Jn(ρ) (thin black lines), for orders n 0, 10, 30, 50 and point numbers N 21, 61, 101 . Heavy black lines replace both ∈ { } 16 ∈ { } where the ‘discrete’ and the ‘continuous’ Bessel functions differ by less than 10− . Following Miller [1], the phase in front of (12), The distinction between even and odd cases of n, as done in(θ +π/2) n 2πink/N n (N) e k = i e = i √NΦn (θk), is extracted to in [3], is subtle but importantto obtain the correct result for write the functions as all n’s (cf.[2, Eq.