A Bernstein--Ganzburg limit theorem for best weighted approximation

Abstract: We prove a Bernstein--Ganzburg type limit relation [ \lim_{n\to\infty} \Bigl(\frac{n}σ\Bigr)^{(2a+1)/p}E_{n,σ}(f)_{p,a,b} =A_σ(f){p,a}, ] where $E{n,σ}(f){p,a,b}$ is the error of best approximation of $f(nt/σ)$ by trigonometric polynomials of degree at most $n$ in $L^{p}((-π,π],|2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}\,dt)$, and $Aσ(f){p,a}$ is the error of best approximation of $f$ by entire functions of exponential type at most $σ$ in $L^{p}(\mathbb{R},|x|^{2a}\,dx)$. For $a=b=0$, this result was obtained by M.~I.~Ganzburg. The proof uses ideas from the Bernstein--Ganzburg limit theorems and a localization method with the Fejér kernel from the proof of the limit relation for Nikol'skii constants. As an application, using known results for polynomial approximation, we compute the exact value of $Aπ(\mathbf{1}{(-1,1)}){1,a}$ for $a=0$ and $a=1/2$.

• The paper establishes a limit relation connecting best weighted approximations by periodic trigonometric polynomials and entire functions.
• It employs advanced localization and kernel methods to relate periodic Jacobi weights to polynomial weights on ℝ and derive asymptotic error bounds.
• The result extends classical unweighted Bernstein–Ganzburg theorems and computes sharp constants for weighted L^p approximation problems.

Bernstein–Ganzburg-Type Limit Theorem for Best Weighted Approximation

Abstract and Motivation

The paper "A Bernstein–Ganzburg limit theorem for best weighted approximation" (2607.02669) establishes a sharp limit relation connecting the errors in best weighted approximation of functions by periodic trigonometric polynomials and entire functions of exponential type. This result generalizes classical Bernstein–Ganzburg-type limit theorems to the setting of $L^p$ spaces under Jacobi-type weights, and extends previous unweighted limits to a broader class of analytic weights. The connection enables transfer of asymptotic error bounds between periodic and entire settings—critical for understanding the precise structure of approximation in weighted environments.

Statement of Main Result

Let $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$ ($a\geq b \geq 0$) define the periodic Jacobi weight on $(-\pi,\pi]$, and consider $f \in L^p(\mathbb{R},|x|^{2a}\,dx)$ for $1 \leq p < \infty$. Write $\mathcal{T}_n$ for the space of trigonometric polynomials of degree at most $n$ and $\mathcal{E}_\sigma$ for the set of entire functions of exponential type at most $\sigma$. The central limit relation proven is

$w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$0

where $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$1 is the best weighted $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$2 approximation error by trigonometric polynomials, and $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$3 is the error associated with best $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$4 approximation by entire functions of given type with the same (now algebraic) weight. For $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$5, this recovers the result of Ganzburg, but the general weighted case is new.

Technical Approach and Proof Outline

The proof employs advanced localization and kernel methods, echoing techniques from the study of Nikol'skii constants but adapting them to the periodic weighted case. Central tools include:

• Local Scaling of Jacobi Weights: For fixed $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$6, the rescaled periodic Jacobi weight converges to the polynomial weight, i.e., $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$7 as $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$8. This underpins the reduction from the periodic case to the entire line.
• Fejér Kernel Localization: The Fejér kernel $w_{a,b}(t) = |2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}$9 is utilized to control the local behavior and regularization of trigonometric polynomials, ensuring asymptotic matching with entire functions. The smoothing function $a\geq b \geq 0$0 is constructed for tight control of the approximation and manages the spread in the periodic to entire limit transition.
• Two-Stage Limiting Argument: The error is first estimated for a general scaling parameter $a\geq b \geq 0$1, then shown to be insensitive as $a\geq b \geq 0$2 due to the uniformity of the weight convergence.
• Error Decomposition: Through careful splitting of the error—central region, kernel tail, and polynomial approximation—a precise upper and lower bound is established, both asymptotically tight, leading directly to the limit theorem.

Explicit Applications

The paper applies the main theorem to compute sharp constants in classical weighted approximation scenarios:

• For the characteristic function $a\geq b \geq 0$3 in the unweighted case ($a\geq b \geq 0$4), the sharp constant is

$a\geq b \geq 0$5

where $a\geq b \geq 0$6 solves $a\geq b \geq 0$7.

• For the "sine" weighted case ($a\geq b \geq 0$8), it is shown that

$a\geq b \geq 0$9

via algebraic polynomial approximation and explicit calculation using change of variables.

These calculations are realized by importing earlier results from polynomial and trigonometric approximation literature, facilitated by the established limit relation. Notably, for $(-\pi,\pi]$0, the asymptotic constant coincides with the natural $(-\pi,\pi]$1 error in weighted polynomial settings.

Implications, Significance, and Prospects

This extension of Bernstein–Ganzburg-type theorems to weighted periodic settings provides a powerful bridge between polynomial approximation on intervals and entire function approximation on the real line—each with their respective natural weights. The result gives a rigorous route to transfer sharp constants and extremal function properties between highly structured spaces. This is significant for the precise study of spectral approximation, best constants in inequalities (e.g., Markov, Bernstein, Nikol'skii), and the analysis of signal recovery in weighted norms.

In theoretical approximation theory, the theorem enriches the repertoire of equivalence principles between periodic and real-line settings under general weights. Practically, it enables computation of weighted best-approximation errors in settings where direct analysis would be cumbersome. The explicit calculation of $(-\pi,\pi]$2 and $(-\pi,\pi]$3 highlights the utility of the theorem in concrete problems.

A more ambitious extension would be to characterize $(-\pi,\pi]$4 for all $(-\pi,\pi]$5, especially for $(-\pi,\pi]$6 not corresponding to known explicit one-sided polynomial approximation formulas. This remains open and challenging except for some special $(-\pi,\pi]$7; existing literature on weighted one-sided approximation provides results in related but distinct settings.

Conclusion

The paper delivers a general and technically rigorous Bernstein–Ganzburg-type limit theorem for best weighted $(-\pi,\pi]$8 approximation by trigonometric polynomials in periodic weighted spaces, covering a broad class of Jacobi-type weights and connecting them asymptotically with entire function approximation. The findings not only extend classical results but also enable sharp quantitative analysis in weighted approximation, providing tools relevant for further exploration in both theoretical and applied analysis.