Logarithmic-Sinc Approximation

• Logarithmic-Sinc approximation is a technique that fuses classical Sinc expansions with logarithmic mappings to expand analyticity regions and boost convergence rates.
• The method replaces traditional conformal maps with logarithmic or logarithmically-modified meshes to refine error bounds and improve efficiency on semi-infinite or infinite intervals.
• It also forms the basis for advanced double-exponential refinements, making it a powerful tool in high-precision approximations for differential equations and integral evaluations.

Logarithmic-Sinc approximation denotes a class of Sinc approximation, interpolation, and differentiation schemes in which the classical Sinc expansion on $\mathbb R$ is combined with a logarithmic change of variables, most prominently $t=\log(1+e^x)$, or with logarithmic structures embedded in more elaborate conformal maps. In the cited literature, the term is not fully standardized: some papers treat it as a logarithmic conformal-map variant of Sinc approximation on semi-infinite or infinite intervals, while related work uses logarithmically modified mesh selection or places a logarithmic map inside a double-exponential transformation. Across these variants, the common objective is to enlarge the admissible strip of analyticity of the transformed function, improve the convergence constant in root-exponential estimates, or, in double-exponential settings, attain almost-exponential convergence (Okayama et al., 2018, Okayama et al., 28 Mar 2025, Okayama, 13 Oct 2025, Sytnyk, 2018).

1. Terminology and scope

The designation “logarithmic-Sinc approximation” does not refer to a single canonical algorithm in the available literature. One established usage concerns Sinc approximation on $(0,\infty)$ or $(-\infty,\infty)$ after replacing older conformal maps by logarithmic maps such as

$t=\phi(x)=\log(1+e^x),$

or by logarithmic composites such as

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$

with the explicit aim of widening the analyticity strip of the transformed function and improving the convergence rate constant (Okayama et al., 2018, Okayama et al., 28 Mar 2025).

A second, related usage appears when logarithmic dependence enters through parameter selection rather than through the map itself. In the extension of Sinc interpolation to algebraically decaying functions, the optimal mesh width is expressed through the Lambert-$W$ function, and a simplified practical choice replaces $W$ by $\ln$, producing a logarithmically modified Sinc discretization rather than a new basis (Sytnyk, 2018).

Several nearby topics are distinct and should not be conflated with this family. “An analog of the Sinc approximation for periodic functions” proposes an interpolation formula for periodic functions and reports exponential convergence for analytic periodic functions, but the available metadata do not describe a logarithmic map (Ogata, 2019). “Numerical Approximation of the logarithmic Laplacian via sinc-basis” uses shifted and dilated sinc functions for a Fourier-multiplier operator with symbol $\log(|\omega|^2)$; there, “logarithmic” refers to the operator, not to a logarithmic conformal map (Dondl et al., 15 Sep 2025). Likewise, incomplete cosine expansions of the sinc function generate rational approximations for the sinc function or the complex error function, but they are not conformal-map-based logarithmic-Sinc methods (Abrarov et al., 2014, Abrarov et al., 2018).

2. Classical Sinc framework and the logarithmic map on $t=\log(1+e^x)$0

The underlying approximation framework is the standard Sinc expansion on the real line,

$t=\log(1+e^x)$1

followed by a conformal pullback from a physical interval to $t=\log(1+e^x)$2. For functions $t=\log(1+e^x)$3 on $t=\log(1+e^x)$4, one introduces a map $t=\log(1+e^x)$5 or $t=\log(1+e^x)$6, defines the transformed function $t=\log(1+e^x)$7, and writes

$t=\log(1+e^x)$8

In this setting, the classical map is Stenger’s

$t=\log(1+e^x)$9

whereas the logarithmic replacement is

$(0,\infty)$0

Both maps send $(0,\infty)$1 onto $(0,\infty)$2, but they differ in complex-analytic continuation properties (Okayama et al., 2018).

The principal theoretical gain is geometric. For the transformed function $(0,\infty)$3, analytic continuation is limited by singularities of $(0,\infty)$4 at $(0,\infty)$5, so the admissible strip satisfies $(0,\infty)$6. For the logarithmic map, $(0,\infty)$7 is not analytic at $(0,\infty)$8, and the transformed function can be analytic in

$(0,\infty)$9

for any $(-\infty,\infty)$0. Under the decay condition

$(-\infty,\infty)$1

the error retains the form

$(-\infty,\infty)$2

with $(-\infty,\infty)$3, but the larger allowable $(-\infty,\infty)$4 improves the effective convergence rate because the exponent depends on $(-\infty,\infty)$5 (Okayama et al., 2018).

This formulation remains a Sinc approximation in the classical sense: the basis is unchanged, and the gain comes from a logarithmic conformal map that better matches the analyticity geometry and the endpoint/decay structure of semi-infinite problems. The literature therefore treats the method as a logarithmic conformal-map enhancement of Sinc approximation rather than as a distinct Sinc basis (Okayama et al., 2018).

3. Logarithmic-Sinc differentiation on infinite intervals

The same design principle extends from function approximation to differentiated Sinc formulas on infinite intervals. In Stenger’s framework, derivatives are approximated by

$(-\infty,\infty)$6

where the weight $(-\infty,\infty)$7 compensates for endpoint singular behavior in derivatives of $(-\infty,\infty)$8. The 2025 refinement keeps this differentiated Sinc construction and replaces the original conformal maps in the logarithmic cases by new logarithmic maps (Okayama et al., 28 Mar 2025).

For $(-\infty,\infty)$9, the original map is

$t=\phi(x)=\log(1+e^x),$0

and the new map is

$t=\phi(x)=\log(1+e^x),$1

For $t=\phi(x)=\log(1+e^x),$2, the original map is

$t=\phi(x)=\log(1+e^x),$3

and the new logarithmic map is

$t=\phi(x)=\log(1+e^x),$4

In both cases, the asymptotic estimate remains root-exponential,

$t=\phi(x)=\log(1+e^x),$5

but the admissible strip expands from $t=\phi(x)=\log(1+e^x),$6 to $t=\phi(x)=\log(1+e^x),$7, so the exponential constant improves (Okayama et al., 28 Mar 2025).

The proof mechanism is explicit. The error is split into a discretization part and a truncation part. The discretization term is controlled by a Hardy-type strip estimate involving

$t=\phi(x)=\log(1+e^x),$8

while the truncation term is bounded by exponential tail estimates. Choosing

$t=\phi(x)=\log(1+e^x),$9

balances the two terms and yields the final root-exponential rate. The improvement is therefore not a change of asymptotic type but a change in the analyticity geometry that enlarges $t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$0 and sharpens the constant in the exponent (Okayama et al., 28 Mar 2025).

4. From logarithmic maps to double-exponential refinements

Within the same variable-transform Sinc tradition, logarithmic maps also serve as intermediates between single-exponential and double-exponential constructions. For unilateral rapidly decreasing functions on $t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$1, Stenger’s transformation

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$2

yields root-exponential convergence,

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$3

A later logarithmic-type refinement,

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$4

improves constants and the admissible analyticity/decay parameters, but still remains in the same qualitative regime,

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$5

The subsequent double-exponential map

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$6

changes the convergence class to

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$7

more precisely

$t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$8

Here the logarithmic structure is retained, but an inner $t=2\sinh\!\bigl(\log(\log(1+e^x))\bigr),$9 turns the construction into a double-exponential map (Okayama, 13 Oct 2025).

A closely related framework appears in Sinc-collocation for initial value problems with exponential-decay end behavior. There, the single-exponential transformation is

$W$0

and the double-exponential transformation is

$W$1

The reported rates are

$W$2

for the single-exponential setting and

$W$3

for the double-exponential setting. The collocation formulation avoids the sine integral $W$4 in the basis functions, but acquires an extra $W$5 factor in the error bound. This suggests that logarithmic-Sinc methods are best understood as part of a broader hierarchy of transformed Sinc schemes in which logarithmic maps are often the single-exponential baseline and the starting point for more aggressive double-exponential refinements (Okayama et al., 2023).

5. Error mechanisms, analyticity strips, and mesh selection

The error analysis of logarithmic-Sinc approximation is governed by two recurring quantities: the half-width $W$6 of the strip of analyticity

$W$7

and the decay parameters that control truncation of the Sinc series. The standard proof pattern separates the total error into a discretization term, caused by replacing the continuous analytic function by the full infinite Sinc lattice sum, and a truncation term, caused by cutting that lattice to $W$8. In transformed Sinc formulas, enlarging the analyticity strip improves the discretization term, while matching the map to endpoint or tail behavior improves the truncation term (Okayama et al., 2018, Okayama et al., 28 Mar 2025).

For logarithmic conformal-map formulas with exponential or mixed decay, the usual balancing choice is

$W$9

with $W$0. This is the parameter that equalizes the discretization scale $W$1 and the truncation scale $W$2, leading to the root-exponential law $W$3. In the double-exponential setting, by contrast, the balance changes to a near-linear exponent, and step sizes take the form

$W$4

or

$W$5

depending on the precise transformation and problem class (Okayama, 13 Oct 2025, Okayama et al., 2023).

In algebraic-tail problems, the truncation mechanism is no longer exponential. For functions analytic in a strip and satisfying

$W$6

the optimal step size is expressed through the Lambert-$W$7 function,

$W$8

and the resulting convergence is

$W$9

A simplified logarithmic mesh replaces $\ln$0 by $\ln$1, giving

$\ln$2

This is a logarithmically modified Sinc discretization in which the basis remains classical and only the optimal balancing law becomes logarithmic (Sytnyk, 2018).

The general conformal-map viewpoint places these results in a wider numerical-analytic context. If a transformed integrand is analytic in a strip and decays double exponentially on the real axis, Sinc and trapezoidal methods inherit near-optimal

$\ln$3

behavior. Polynomial adjustments to $\ln$4 can be used to move singularities to the strip boundary and maximize the product $\ln$5, thereby sharpening the convergence constant. This broader theory clarifies why logarithmic maps are effective: they are particular instances of analyticity-strip engineering by conformal transformation (Slevinsky et al., 2014).

6. Numerical evidence, practical behavior, and common distinctions

The numerical evidence reported for logarithmic conformal-map variants is consistent across several problem classes. On the semi-infinite interval $\ln$6, the replacement of $\ln$7 by $\ln$8 was tested on

$\ln$9

and the observed maximum errors showed that the new approximation consistently converges faster than the old approximation; the theoretical bounds also enclosed the observed errors (Okayama et al., 2018).

For differentiated Sinc formulas, numerical experiments were reported for $\log(|\omega|^2)$0 and for approximations of $\log(|\omega|^2)$1, $\log(|\omega|^2)$2, and $\log(|\omega|^2)$3. On $\log(|\omega|^2)$4, the test function

$\log(|\omega|^2)$5

was evaluated on $\log(|\omega|^2)$6 points $\log(|\omega|^2)$7, $\log(|\omega|^2)$8, and on $\log(|\omega|^2)$9 the function

$t=\log(1+e^x)$00

was evaluated on $t=\log(1+e^x)$01 points $t=\log(1+e^x)$02, $t=\log(1+e^x)$03, together with $t=\log(1+e^x)$04. In both examples, the improved logarithmic formulas were reported to converge faster than Stenger’s formulas (Okayama et al., 28 Mar 2025).

For unilateral rapidly decreasing functions, the double-exponential logarithmic refinement was tested on

$t=\log(1+e^x)$05

with errors measured on $t=\log(1+e^x)$06 points $t=\log(1+e^x)$07, $t=\log(1+e^x)$08, plus $t=\log(1+e^x)$09. The new DE-Sinc method converged faster than the methods based on $t=\log(1+e^x)$10 and $t=\log(1+e^x)$11, and the observed errors lay below the predicted error bounds (Okayama, 13 Oct 2025).

A recurrent misconception is that every appearance of “logarithmic” and “sinc” denotes the same methodology. The literature instead separates at least three strands: logarithmic conformal-map Sinc approximation on unbounded intervals; sinc-basis discretization of operators such as the logarithmic Laplacian, where the symbol is $t=\log(1+e^x)$12; and sampling-based rational approximations derived from incomplete cosine expansions of the sinc function. The first strand is the one most directly described by the expression logarithmic-Sinc approximation. The others are methodologically adjacent, but they solve different approximation problems and use different analytical mechanisms (Dondl et al., 15 Sep 2025, Abrarov et al., 2018, Abrarov et al., 2014).