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Discrete Kontorovich-Lebedev transforms

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Discrete analogs of the Kontorovich-Lebedev transform are introduced, with expansions for sequences and functions in Macdonald and incomplete Bessel functions, and an application to the Helmholtz equation.

desk verdict New discrete Kontorovich–Lebedev expansions with real potential, but Theorem 7's second part as stated is false—the proof proves a different expansion. read the letter →

arxiv 1908.01392 v2 pith:QCNIYR5L submitted 2019-08-04 math.CA

Semyon Yakubovich This is my paper · ORCID
classification math.CA MSC 45A0544A1542A1633C10
keywords Kontorovich-LebedevtransformmodifiedBesselfunctionMacdonaldincompleteFourierseriesDirichletproblemHelmholtzequationindex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces discrete analogs of the classical Kontorovich-Lebedev transform, replacing the continuous index integral by series over Macdonald functions $K_{in}(x)$ paired with incomplete Bessel functions $J(x,in,\pi)$, $K_c(x,in,\pi)$, and $K_s(n,i\tau,\sinh^{-1}(\pi))$. It establishes biorthogonality of these kernel sequences under suitable measures and proves a family of expansions, formulas (2.30)-(2.41), recovering a sequence term $a_n$ or a function $f(x)$ from such series and integrals. A sympathetic reader would care because these formulas give a genuinely discrete, Fourier-style calculus for the index transform, and the paper demonstrates the payoff by solving a Dirichlet boundary-value problem for the inhomogeneous Helmholtz equation in the upper half-plane explicitly.

What carries the argument

The load-bearing objects are the incomplete modified Bessel function $J(z,\nu,w)=\int_0^w e^{-z\cosh u}\cosh(\nu u)\,du$ and its trigonometric siblings $K_c(x,\nu,w)$ and $K_s(x,\nu,w)$, defined by cutting the standard integral representations (1.3)-(1.4) at $w$. These sit between the full Macdonald function $K_\nu(z)$ and Fourier series: cutting the integrals makes $u\in[0,\pi]$ a natural domain on which the Fourier-series identities (2.9)-(2.13) hold pointwise. The biorthogonality relations (2.22)-(2.24), obtained by combining those Fourier expansions with classical integral evaluations (2.26), (2.27), and (2.29), are what turn the series-and-integral expressions into inversion formulas.

What would settle it

Take the sequence $a_m=e^{-m}$, which satisfies the paper's decay conditions, and numerically evaluate both sides of expansion (2.35) for $n=1$ and a fixed $x>0$: if the improper integral over $\tau$ of $\tau\sinh(\pi\tau)K_{i\tau}(1)\sum_m a_m K_s(m,i\tau,\sinh^{-1}\pi)$ does not converge to $a_1$, the claimed inversion fails; a similar check of the limit $\alpha\to\pi/2^-$ in (2.34) would settle the Abel-regularized case.

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Extended reading notes

Core claim

The central claim is that the classical Kontorovich-Lebedev inversion can be discretized: for suitable sequences $\{a_n\}$ and functions $f$, formulas (2.30)-(2.35) recover $a_n$ from integrals against $\sum_m a_m K_{im}(x)$ or $\sum_m a_m J(x,im,\pi)$, and formulas (2.36)-(2.41) recover $f(x)$ from series over $J(x,in,\pi)$, $K_c(x,in,\pi)$, $K_{ix}(n)$, or $K_s(n,ix,\sinh^{-1}\pi)$. The proof rests on biorthogonality of the kernel sequences $\{K_{in}(x)\}$ with $\{J(x,in,\pi)\}$ and $\{K_c(x,in,\pi)\}$, and $\{K_{i\tau}(n)\}$ with $\{K_s(n,i\tau,\sinh^{-1}\pi)\}$, with respect to $dx/x$ and $\tau\sinh(\pi\tau)\,d\tau$. As an application, the paper shows that a specific series in $J(r,in,\pi)$ solves the inhomogeneous Helmholtz equation $\Delta u-u=h$ in the upper half-plane, vanishing at infinity and taking prescribed boundary values $u(r,0)=0$, $u(r,\pi)=f(r)$.

Load-bearing premise

The expansions proved via Abel regularization depend on being allowed to exchange the limit with infinite sums and integrals; the paper leaves some of those exchanges to the reader or asserts them by uniform convergence without a fully detailed verification.

Editorial extensions

If this is right

  • A sequence $(a_n)$ satisfying $\sum |a_n|e^{-\pi n/2}<\infty$ can be recovered from the function $f(x)=\sum_m a_m K_{im}(x)$ through the integral formula (2.30), and dually, under $\sum |a_n|/n<\infty$, through (2.31).
  • Functions represented as $f(x)=\int_{-\pi}^{\pi} e^{-x\cosh u}\phi(u)\,du$ with Lipschitz $\psi$ admit the expansion (2.36) in the incomplete-Bessel kernels $J(x,in,\pi)$.
  • Functions represented as $f(x)=\int_{-\pi}^{\pi}\sin(x\sinh u)\phi(u)\,du$ admit the expansion (2.38) in the kernels $K_c(x,in,\pi)$.
  • The series $u(r,\theta)=\frac{2}{\pi^2}\sum_{n=1}^\infty n\sinh(\theta n)J(r,in,\pi)a_n$ solves the inhomogeneous Helmholtz equation (4.3) in the upper half-plane, vanishes at infinity, and, when $a_n$ is the discrete Kontorovich-Lebedev transform of $f$, satisfies boundary conditions $u(r,0)=0$, $u(r,\pi)=f(r)$.
  • Under stronger decay conditions, expansions (2.34), (2.35), (2.40), and (2.41) provide Abel-regularized and improper-sense inversions involving the $K_s$ kernels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending the paper's method, one could try to construct discrete analogs of other index transforms by replacing continuous index integration with a sum over discrete indices and using finite-window integral representations.
  • The same biorthogonality machinery could support spectral or quadrature methods for Helmholtz-type boundary problems in half-plane and wedge geometries, using expansions (2.36)-(2.41) as approximation bases.
  • A numerical check of the Abel-regularized formulas (2.34) and (2.35) on simple sequences such as $a_m=e^{-m}$ would clarify the exact convergence conditions the paper leaves implicit.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper introduces discrete analogs of the classical Kontorovich-Lebedev transform, based on series involving Macdonald functions K_{in}(x) and incomplete Bessel functions J(x,in,π), Kc(x,in,π), Ks(n,iτ,sinh^{-1}(π)). The main results are expansions (2.30)-(2.41) for sequences and functions, a biorthogonality lemma (Lemma 1), and an application to solving a Dirichlet boundary value problem for the inhomogeneous Helmholtz equation in the upper half-plane. The proofs rely on integral representations, standard integral formulas, and estimates such as Lebedev's inequality.

Significance. If the stated results are correct, the paper provides a useful discrete counterpart of a well-known integral transform, with explicit expansion formulas and a PDE application. The author gives detailed estimates in several proofs, uses machine-checkable standard integral formulas, and proves the biorthogonality relations with nontrivial limiting arguments. However, one main theorem is false as stated, and another expansion is left essentially unproved, so the paper needs substantial revision before it can be accepted.

major comments (5)
  1. [Theorem 4, proof of (2.35)] The second part of Theorem 7, expansion (2.41), is false as stated. The theorem says f is given by series (3.3), i.e. f(x)=∑_{m=1}^∞ a_m K_{im}(x), but the proof substitutes f(τ)=∑_{m=1}^∞ a_m K_{iτ}(m). These are different functions: in the first the index is the discrete integer m and the argument is x; in the second the index is iτ and the argument is m. The proof establishes (2.41) for the corrected class f(x)=∑_{m=1}^∞ a_m K_{i x}(m). For the stated class, take a_1=1 and all other a_m=0, so f(τ)=K_i(τ). As τ→∞, K_i(τ)∼(π/(2τ))^{1/2}e^{-τ}, while by (3.10) K_s(1,iτ,sinh^{-1}(π))=O(e^{-πτ/2}/τ^2). Thus the integrand in (2.41) behaves like C τ^{-3/2} e^{(π/2-1)τ}, which diverges; the Abel sense (2.34) does not rescue it because the factor cosh(ατ) makes the divergence worse for α>1. The statement and proof must be reconciled, and the theorem should be corrected to the class for which the proof actually works.
  2. [Theorem 7, proof of (2.40)-(2.41)] Expansion (2.35) is one of the main discrete analogues, but its proof is explicitly left to the reader: after invoking biorthogonality (2.24) and (2.28), the author states only that the order of integration and summation is justified and then writes "The completion of the proof is left to the reader." This is a load-bearing gap, because (2.35) involves an improper integral over τ, an infinite series, and the interplay of the limiting procedures used in the proof of Lemma 1. A rigorous verification of these interchanges, with explicit estimates, is necessary to establish (2.35).
  3. [Theorem 4, statement] In the proof of the first part of Theorem 7 and in the (intended) proof of (2.41), the author asserts that "all interchanges of the summation, integration and the passage to the limit are allowed via the absolute and uniform convergence," but no uniform bounds are provided. In particular, in the chain leading to (2.41) one must justify exchanging the outer sum over n, the inner series over m, the improper τ-integral, and the Abel limit α→π/2−. These are nontrivial because the kernel Ks and the Macdonald functions have oscillatory and exponential behavior; the earlier proof of a similar interchange in Lemma 1 required a detailed asymptotic argument. Without explicit verification, the expansions (2.40) and (2.41) are not fully established.
  4. [Theorem 7 proof, summation index] The statement of Theorem 4 says the general term can be expanded "with respect to (2.33)" with convergence in the Abel sense, but the proof establishes (2.34). This appears to be a typo: it should refer to (2.34). While not a technical error by itself, it confuses which expansion is being proved and should be fixed.
  5. [Abstract] In the proof of Theorem 7, after Eq. (3.12), the same symbol n is used for the outer summation index and for the inner summation index in expressions such as "∑_{n=1}^∞ a_m"; this should be m. The repeated use of n obscures the argument and should be corrected.
minor comments (5)
  1. [Introduction, displayed formulas] The abstract contains a spacing typo: "tra nsforms" should be "transforms".
  2. [Section 2, Eq. (2.24)] In Eq. (1.1), the factor 2/(π^2 τ sinh(πτ)) is written without a space; the typesetting is otherwise clear, but this should be cleaned up to avoid ambiguity between τ sinh(πτ) and τ·sinh(πτ).
  3. [Section 3, Theorem 4] The proof of Lemma 1 uses the value of the improper integral (2.29) from reference [4], but the passage to the limit under the integral in (2.28) is justified with an asymptotic argument; the wording "The problem now is to motivate..." is somewhat informal and should be rephrased to state directly that the justification is provided in the following lines.
  4. [Section 4, Eq. (4.5)] In the proof of Theorem 4, the estimate ∫_0^∞ cosh(ατ)/√(sinh(πτ)) dτ < ∞ for 0<α<π/2 is asserted without a reference; a one-line justification would be helpful.
  5. [References] In Theorem 8, the function u(r,θ) is written as u(r,θ)=2/π^2 ∑ n sinh(θn) J(r,in,π) a_n; this is consistent with the text, but in the boundary condition (4.7) the notation f(r) is used both for the boundary data and for the function in Theorem 5; this is not a mathematical error but could be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the discrete expansions are derived from the paper's own biorthogonality lemmas and from standard, independently stated integral formulas; the author's self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. Each expansion (2.30)-(2.41) is proved by substituting the stated integral representations (2.7)-(2.8), interchanging sums and integrals under explicit convergence conditions, and then applying the biorthogonality identities of Lemma 1, which are themselves derived from standard integral evaluations (2.26), (2.27), and (2.29). No expansion is assumed as an input; the target coefficient a_n appears only after Fourier orthogonality of sin(nu) sin(mu). The cited results from the author's monograph [4] (Lebedev inequality, asymptotics, formula (2.29)) are parameter-free classical facts that do not contain the discrete expansions being proved, so they do not make the argument circular. Two non-circular caveats are flagged: Theorem 4 explicitly leaves the proof of (2.35) to the reader, and Theorem 7's proof of (2.41) appears to substitute f(tau)=sum a_m K_{i tau}(m) while the stated class (3.3) uses K_{i m}(x); these are completeness and correctness issues, not self-referential derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard facts in special function theory (known integrals from Prudnikov et al., Lebedev's inequality, asymptotics of Macdonald functions) and on Fourier series theory. The only 'hand-chosen' parameters are the upper limits w=π and w=sinh^{-1}(π) in the incomplete Bessel functions, which are selected to make the Fourier biorthogonality work. No empirical fitting or invented entities are involved.

free parameters (2)
  • Upper integration limit π in incomplete Bessel functions J(x,i n,π), Kc(x,i n,π) = π
    The truncation of the integral representations at w=π is chosen so that the resulting functions are biorthogonal to {K_{i n}(x)} with respect to the Fourier sine/cosine basis on [0,π]; this choice is not forced by the problem and is essential to the expansions.
  • Upper integration limit sinh^{-1}(π) in Ks(n,i τ, sinh^{-1}(π)) = sinh^{-1}(π) ≈ 1.862...
    For the sequence {K_{i τ}(n)}_n and {Ks(n,i τ, sinh^{-1}(π))}_n biorthogonality, the upper limit is set to sinh^{-1}(π) to match the substitution u = sinh(t) and the interval [0,π]; this is an ad hoc choice.
assumptions (4)
  • standard math Known integral formulas (2.26), (2.27), (2.29), (3.12) from Prudnikov et al. are correct
    These external integrals are used in the proofs of Lemma 1 and Theorems 4 and 7; the paper does not re-derive them.
  • standard math Lebedev inequality (1.9) and the asymptotic of K_{iτ}(n) as τ→∞ are assumed
    Used to justify interchanges and the limit passage in Lemma 1 and Theorem 1; these are standard estimates for the Macdonald function.
  • standard math Fourier series theory, including convergence of sine/cosine series for Lipschitz functions, holds
    Used in Theorems 5 and 6 to prove expansions (2.36) and (2.38).
  • standard math The dominated convergence theorem and Fubini's theorem apply under the stated integrability conditions
    The proofs rely on the ability to interchange summation, integration, and limits; the paper gives estimates to justify these but sometimes asserts them tersely.

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Cite this review

Pith. "Pith review of Discrete Kontorovich-Lebedev transforms." pith.science (2026). https://pith.science/paper/QCNIYR5L

@misc{pith2026190801392,
  author       = {Pith},
  title        = {Pith review of: Discrete Kontorovich-Lebedev transforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCNIYR5L}},
  note         = {Machine review of arXiv:1908.01392}
}
abstract

Discrete analogs of the classical Kontorovich-Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function $K_{in}(x), x >0, n \in \mathbb{N}, i $ is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application, a Dirichlet boundary value problem in the upper half-plane for inhomogeneous Helmholtz equation is solved.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    Jones, Incomplete Bessel functions

    D.S. Jones, Incomplete Bessel functions. I, Proc. Edinb. Math. Soc. 50 (2007), N 1, 173-183

    work page 2007

  2. [2]

    Lebedev, Sur un formule d’inversion, C.R

    N.N. Lebedev, Sur un formule d’inversion, C.R. (Doklady) Acad. Sci. URSS (N.S.) 52 (1946), 655-658 (in French)

    work page 1946

  3. [3]

    Prudnikov, Yu.A

    A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series . Vol. I: Elementary Functions , Vol. II: Special Functions, Gordon and Breach, New York and London, 1986, Vol. III : More special functions , Gordon and Breach, New York and London, 1990

    work page 1986

  4. [4]

    Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996

    S. Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996. S.Yakubovich Department of Mathematics, Faculty of Sciences, University of Porto, Campo Alegre st., 687 4169-007 Porto Portugal E-Mail: syakubov@fc.up.pt

    work page 1996

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