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arxiv: 2005.06076 · v1 · submitted 2020-05-12 · 🧮 math-ph · math.MP

Discrete Bessel functions and transform

Kenan Uriostegui , Kurt Bernardo Wolf This is my paper

Pith reviewed 2026-05-24 14:05 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords discrete Bessel functionsGraf's formulasBessel function discretizationN-point Fourier transformfinite signal analysisproduct-displacement formulasdiscrete special functions
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The pith

Restricting the Fourier transform to N points produces discrete Bessel functions that obey exact analogues of continuous relations including Graf's formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs discrete Bessel functions B^{(N)}_n(x_m) by restricting the Fourier transform in the Bessel generating function to N points on the circle. These functions obey the same linear and quadratic identities as the ordinary Bessel functions J_n(x), notably Graf's product-displacement formulas. A sympathetic reader would care because this supplies an exact N-point transform useful for the Fourier analysis of finite decaying signals, with close numerical agreement to the continuous case whenever n + |m| < N.

Core claim

By discretizing the Bessel integral generating function through an N-point restriction of the Fourier transform over the circle, the resulting discrete Bessel functions B^{(N)}_n(x_m) satisfy several linear and quadratic relations that are exact analogues of those for the continuous Bessel functions, including Graf's formulas, and they approximate the continuous values closely when n + |m| < N, thereby providing an efficient N-point transform between functions of order and position for finite signals.

What carries the argument

The N-point restriction of the Fourier transform in the Bessel integral generating function that defines the discrete Bessel functions B^{(N)}_n(x_m).

If this is right

  • The discrete Bessel functions satisfy exact analogues of linear and quadratic relations including Graf's product-displacement formulas.
  • They approximate continuous Bessel functions closely for n + |m| < N.
  • This provides an N-point transform between functions of order and of position for finite decaying signals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method of restricting the Fourier transform could apply to discretizing other special functions with similar integral representations.
  • The exact discrete relations may enable new finite-domain numerical schemes for problems involving cylindrical wave expansions.
  • Close approximation in the central range suggests utility for signal processing on finite lattices with Bessel-like symmetries.

Load-bearing premise

Restricting the Fourier transform over the circle to N points preserves the generating properties and yields exact analogues of the continuous Bessel relations.

What would settle it

Direct computation for small N of whether Graf's product-displacement formula holds exactly for the discrete functions, or measurement of how closely they approximate the continuous Bessel values outside n + |m| < N.

Figures

Figures reproduced from arXiv: 2005.06076 by Kenan Uriostegui, Kurt Bernardo Wolf.

Figure 1
Figure 1. Figure 1: Comparison of values of the discrete Bessel functions [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the left-hand sides of Eqs. (23) and (24) for integer [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of values of the discrete and continuous Bessel func [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

We present a straightforward discretization of the Bessel functions $J_n(x)$ to discrete counterparts $B^{(N)}_n(x_m)$, of $N$ integer orders $n$ on $N$ integer points $x_m \equiv m$, that we call discrete Bessel functions. These are built from a Bessel integral generating function, restricting the Fourier transform over the circle to $N$ points. We show that the discrete Bessel functions satisfy several linear and quadratic relations, particularly Graf's product-displacement formulas, that are exact analogues of well-known relations between the continuous functions. It is noteworthy that these discrete Bessel functions approximate very closely the values of the continuous functions in ranges $n + |m| < N$. For fixed $N$, this provides an $N$-point transform between functions of order and of position,$f_n$ and $\widetilde{f}_m$, which is efficient for the Fourier analysis of finite decaying signals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces discrete Bessel functions B_n^{(N)}(x_m) obtained by replacing the continuous Fourier integral in the generating function for J_n(x) with an N-point discrete sum over roots of unity. It asserts that these functions obey exact linear recurrences and quadratic identities that are direct analogues of the continuous case (in particular Graf's addition formulas), that they approximate J_n(m) closely for n + |m| < N, and that the construction yields an efficient N-point transform for finite decaying signals.

Significance. If the exact quadratic identities are shown to hold, the construction would supply a parameter-free (apart from the single integer N) discrete analogue of Bessel functions with potential utility in finite-signal Fourier analysis. The exactness of the analogues, rather than mere numerical closeness, would be the principal strength; the approximation statement would also benefit from quantitative error bounds.

major comments (1)
  1. [Abstract] Abstract (construction paragraph): the claim that the N-point discretization preserves exact analogues of Graf's product-displacement formulas is load-bearing for the central contribution, yet the provided text supplies neither a derivation nor an explicit verification. The finite-sum definition immediately yields linear relations via geometric summation, but the quadratic identities rely on orthogonality properties that hold only for band-limited functions on the circle; nothing in the sampling step automatically guarantees that the product of two discrete sums equals the required single sum, so an explicit algebraic check or counter-example for small N is required.
minor comments (2)
  1. The abstract states that 'several linear and quadratic relations' are satisfied but does not exhibit even one explicit formula; inserting at least the discrete version of Graf's formula (with the corresponding continuous reference) would make the claim concrete.
  2. Notation: the identification x_m ≡ m should be stated explicitly (including any scaling factor) and the range of the index m clarified relative to N.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (construction paragraph): the claim that the N-point discretization preserves exact analogues of Graf's product-displacement formulas is load-bearing for the central contribution, yet the provided text supplies neither a derivation nor an explicit verification. The finite-sum definition immediately yields linear relations via geometric summation, but the quadratic identities rely on orthogonality properties that hold only for band-limited functions on the circle; nothing in the sampling step automatically guarantees that the product of two discrete sums equals the required single sum, so an explicit algebraic check or counter-example for small N is required.

    Authors: We agree that the abstract itself contains no derivation or explicit verification of the quadratic identities, which is a valid observation. The full manuscript derives these exact analogues of Graf's formulas in Section 3 by direct algebraic manipulation of the finite sums over the N-th roots of unity. Because the discrete Bessel functions are defined as the coefficients in the N-point discrete Fourier expansion of the generating function, the product of two such expansions reduces exactly to a single sum via the discrete orthogonality relation sum_{k=0}^{N-1} exp(2 pi i k (l-m)/N) = N delta_{l m mod N}. This identity holds by construction on the cyclic group without requiring continuous band-limited assumptions. We will revise the manuscript to include an explicit algebraic verification for small N (such as N=4) and will update the abstract to reference Section 3 for the proof of the quadratic identities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines discrete Bessel functions B_n^{(N)}(m) directly as the N-point DFT sampling of the standard Bessel integral generating function. Linear recurrences are derived from finite geometric sum identities, which are independent of the target relations. Quadratic identities such as Graf's formulas are asserted to hold exactly and are shown via the discrete construction; this is a mathematical claim about the defined objects rather than a reduction of the result to its own inputs by definition or fitting. No self-citations, ansatzes, or uniqueness theorems from prior author work are invoked as load-bearing steps in the provided text. The derivation chain remains independent of the claimed relations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central construction depends on the standard properties of the Bessel integral generating function and the discrete Fourier transform on the circle; N is introduced as the grid size without independent calibration.

free parameters (1)
  • N
    The number of discretization points and orders; chosen as the finite grid size that defines the discrete functions.
axioms (1)
  • domain assumption Restricting the Fourier transform over the circle to N points preserves the generating function properties of the continuous Bessel integral.
    Invoked to define B_n^{(N)}(x_m) from the continuous generating function.
invented entities (1)
  • Discrete Bessel function B_n^{(N)}(x_m) no independent evidence
    purpose: To serve as the finite-grid analogue of the continuous Bessel function J_n(x) for use in N-point transforms.
    New object introduced by the restriction procedure; no independent falsifiable evidence supplied beyond the construction itself.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Discrete Bessel and Mathieu functions

    math-ph 2021-02 unverdicted novelty 5.0

    Discrete Bessel and Mathieu functions are introduced as N-point Fourier-sum approximants to the continuous special functions via separation of variables under discrete dihedral symmetry.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · cited by 1 Pith paper

  1. [1]

    Gradshteyn and I.M

    I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Prod- ucts, A. Jeffrey and D. Zwillinger Eds, (Academic Press, 2007)

    work page 2007

  2. [2]

    Biagetti, P

    G. Biagetti, P. Crippa, L. Falaschetti, and C. Turchetti, Discrete Bessel functions for representing the Class of Finite Duration Decaying Se- quences, European Signal Analysis Conference, p. 2126–2130 (Budapest, 2016)

    work page 2016

  3. [3]

    Erd´ elyiet al., Higher Transcendental Functions (Based on notes by H

    A. Erd´ elyiet al., Higher Transcendental Functions (Based on notes by H. Bateman) Vol. 2 (McGraw-Hill, New York, 1953)

    work page 1953

  4. [4]

    Boyer, Discrete Bessel functions, J

    R.H. Boyer, Discrete Bessel functions, J. Math. Anal. Appl. 2, 509–524 (1961)

    work page 1961

  5. [5]

    Bohner and T

    M. Bohner and T. Cuchta, The Bessel difference equation, Proc. Amer. Math. Soc. 145, 1567–1580 (2017)

    work page 2017

  6. [6]

    Slav´ ık, Discrete Bessel functions and partial differential equations,J

    A. Slav´ ık, Discrete Bessel functions and partial differential equations,J. Diff. Eqs. Applics. DOI:10.1080/10236198.2017.141610 (2017)

    work page doi:10.1080/10236198.2017.141610 2017

  7. [7]

    Winternitz, K.B

    P. Winternitz, K.B. Wolf, G.S. Pogosyan, and A.N. Sissakian, Graf’s addition theorem obtained from SO(3) contraction, Theor. Mat. Phys. 129, 1501–1503 (2001)

    work page 2001

  8. [8]

    Watson, Theory of Bessel Functions (Cambridge University Press, 1922)

    G.N. Watson, Theory of Bessel Functions (Cambridge University Press, 1922)

    work page 1922

  9. [9]

    Talman, Special Functions –A Group Theoretic Approach (Based on lectures by E.P

    J.D. Talman, Special Functions –A Group Theoretic Approach (Based on lectures by E.P. Wigner), (W.A. Benjamin Inc., 1968). 11

    work page 1968