Optimal Approximants of Beurling-Selberg Type

• The paper presents explicit extremal entire functions that majorize or minorize target functions under band-limitation constraints in Fourier analysis.
• It utilizes reproducing kernel Hilbert space techniques to solve L² minimization problems and interpolate at prescribed nodes for optimized error bounds.
• Generalizations include interpolation constraints, Gaussian subordination, and multi-dimensional boxes, with applications in analytic number theory and signal processing.

Optimal approximants of Beurling–Selberg type are explicit extremal entire functions that majorize or minorize a given target—most commonly, characteristic functions of intervals (or more general sets), Gaussians, or related kernels—subject to band-limitation constraints and auxiliary optimality criteria. These constructions, rooted in the works of Beurling, Selberg, and Vaaler, are fundamental in Fourier analysis, analytic number theory, and signal processing, where precise control over approximation error, spectral constraints, and interpolation properties is required.

1. Classical Beurling–Selberg Extremal Problems

Given an interval $I=[-a,a]$, the classical Beurling–Selberg problem seeks entire functions (majorant $M$, minorant $m$) of exponential type at most $2\pi\Delta$ such that $m(x)\leq \chi_I(x)\leq M(x)$ for all $x\in\mathbb{R}$ and whose Fourier transforms are supported in $[-\Delta,\Delta]$. The objective is to minimize the $L^1$-error, i.e.,

$$E_+(M) = \int_{\mathbb{R}} \left( M(x) - \chi_I(x) \right) \, dx, \quad E_-(m) = \int_{\mathbb{R}} \left( \chi_I(x) - m(x) \right) \, dx$$

The unique extremal solutions are explicitly given by interpolation at integers or half-integers, and their Fourier transforms exhibit the required compact support. For the interval $[-\delta, \delta]$, Selberg’s extremals are

$$M(x) = \frac{\sin^2(\pi x)}{\pi^2} \left( \sum_{|n+r|\leq\delta} \frac{1}{(x-n-r)^2} + \frac{\lambda_+}{x+\delta} + \frac{\lambda_+}{\delta-x} \right)$$

$$m(x) = \frac{\sin^2(\pi x)}{\pi^2} \left( \sum_{|n+r|<\delta} \frac{1}{(x-n-r)^2} + \frac{\lambda_-}{x+\delta} + \frac{\lambda_-}{\delta-x} \right)$$

for $r \in \{0, \frac{1}{2}\}$ as determined by $\delta \in \mathbb{Z}$ or $\delta \in \mathbb{Z}+\frac{1}{2}$, and suitable residues $\lambda_\pm$ at the endpoints (Mukhametzhanov et al., 2020).

2. Reproducing Kernel and Hilbert Space Methodology

The construction of optimal Beurling–Selberg approximants utilizes reproducing kernels associated with band-limited Paley–Wiener spaces. For support in $[-\delta,\delta]$, the appropriate Hilbert space is $H_\delta$, with reproducing kernel

$$K(\omega,z) = \frac{\sin \pi\delta(z-\overline{\omega})}{\pi(z-\overline{\omega})}$$

An $L^1$-admissible function $F$ with $F(x)\ge0$ and Fourier support in $[-\delta,\delta]$ admits a unique factorization $F(z)=U(z)U^*(z)$ with $U \in H_{\delta/2}$ (Kelly, 2014). The extremal problem then reduces to an $L^2$ minimization: $$\kappa(\alpha,\beta,\delta)=\inf_{U\in H_{\delta/2}} \|U\|_2^2 \quad \text{subject to}\quad \langle U,K(\alpha,\cdot)\rangle \cdot \overline{\langle U,K(\overline\alpha,\cdot)\rangle} = \beta$$ The unique solution corresponds to $U(z)$ lying in the span of the reproducing kernels at $\alpha$ and $\overline\alpha$, with explicit coefficients.

3. Generalizations: Interpolation Constraints and Gaussian Subordination

Extremal approximants can be modified to enforce interpolation constraints, such as vanishing at prescribed points or matching the target at additional nodes. For example, the Carneiro–Littmann variation constructs majorants of $\chi_I$ vanishing at $\alpha\in\mathbb{C}$ by adding an explicit kernel-based correction to the Selberg majorant (Kelly, 2014): $M_\alpha(z) = C(z) + F(z;\alpha,-C(\alpha))$ where $C(z)$ is the classical majorant and $F(z;\alpha,-C(\alpha))$ is the prescribed kernel correction. This constraint increases the $L^1$-mass, typically by $O(\delta^{-2})$ for small $\delta$.

Gaussian subordination further extends the framework to even functions expressible as superpositions of Gaussians $$g(x)=\int_0^\infty e^{-\pi\lambda x^2}d\mu(\lambda)$$, and the optimal majorants/minorants are constructed through integration of the Gaussian approximants against $\mu$ (Carneiro et al., 2010, Carneiro et al., 2015). The minimal error is given by integrating the corresponding error constants: $$\int [g(x)-\ell(x)] dx = \int E^-(\lambda, \tau)\, d\mu(\lambda)$$ This methodology unifies classical cases and yields closed-form optimal constants for a wide range of kernels.

4. High- and Multi-Dimensional Box Problems

For boxes $Q_N=[-1,1]^N$ in high dimensions, one constructs minorants $f$ subject to $\operatorname{supp}\widehat f \subset Q_N$, $f(x) \leq 1_{Q_N}(x)$, maximizing $\int_{\mathbb{R}^N}f(x)\,dx$ (Carruth et al., 2017). Explicit constructions exist for $N\leq5$; for $N\geq718$, no nontrivial minorants with positive mass are possible. These extremal functions take a separated form: $f_N(x) = S_N(x)P_N(x)$ where $S_N(x)$ is a product of 1D Selberg extremals, and $P_N(x)$ is an even polynomial chosen to enforce vanishing at prescribed lattice points. Applications include improved bounds in large-sieve type inequalities.

5. Application Domains: Analytic Number Theory, Modular Bootstrap, Signal Processing

Beurling–Selberg-type approximants underlie sharp inequalities in analytic number theory, such as explicit bounds in zero-density problems and Hilbert-type inequalities (Hejhal, 2013, Carneiro et al., 2010, Carneiro et al., 2015). In the context of modular bootstrap in conformal field theory, extremal majorants/minorants of characteristic functions yield optimal bounds on spectral densities, with explicit dependencies on the band-limit and interval width (Mukhametzhanov et al., 2020). The modular inversion argument relates the extremal integrals directly to bounds on the number of operators in a scaling window. In super-resolution, higher-order box approximants control singular values of Fisher information matrices, determining threshold separations for stable parameter recovery (Costa et al., 2022).

6. Explicit Formulas, Error Constants, and Asymptotics

Extremal functions of Beurling–Selberg type allow for explicit computation of $L^1$-errors, kernel evaluations, and asymptotic rates. In classical cases, these error bounds converge as: $$\int_{\mathbb{R}} (M_{\Delta}(x)-\chi_{I}(x)) dx = \int_{\mathbb{R}} (\chi_{I}(x)-m_{\Delta}(x)) dx = \frac{1}{\Delta}$$ For kernel modifications with interpolation constraints, the error increases: $$\rho(\alpha, I, \delta) \asymp \delta^{-2} \,\, (\delta \rightarrow 0^{+}), \quad \rho(\alpha, I, \delta) \approx \delta^{-1} \,\, (\delta \rightarrow \infty)$$ In multidimensional cases, errors are given by explicit sums over reproducing kernel values at zeros of associated Hermite–Biehler functions (Carneiro et al., 2015): $$U^-_N(2,\lambda) = W_{N-1}\sum_{j} e^{-\pi\lambda\alpha_j^2} K_v(\alpha_j,\alpha_j)$$ with asymptotics controlled by Bessel zero distributions.

7. Open Problems and Extensions

Ongoing challenges include the classification of extremal minorants for boxes in higher dimensions beyond $N=5$, existence and explicit construction of two-sided extremals in multivariate settings, and extension to non-box domains such as balls and simplices. Application-driven extensions involve optimizing for additional constraints, such as higher-order vanishing moments or generalized interpolation at prescribed sets, and exploiting extremal approximants in new analytic and computational frameworks (Carruth et al., 2017, Carneiro et al., 2015).

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Representative Table: Classical Extremal Approximants for $\chi_{[-\delta,\delta]}$

Majorant $M(x)$	Minorant $m(x)$	$L^1$-Error
Explicit Selberg series	Explicit Selberg series	$1/\delta$

All entries correspond to explicit formulas and error constants in the classical 1D case (Hejhal, 2013, Mukhametzhanov et al., 2020).

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The theory of optimal Beurling–Selberg-type approximants is central to harmonic analysis, extremal function theory, and their analytic and computational applications, providing not only explicit constructions but also rigorous bounds and interpolation mechanisms under spectral constraints. Foundational references include Vaaler’s survey (Hejhal, 2013), Carneiro–Littmann’s Gaussian subordination framework (Carneiro et al., 2010, Carneiro et al., 2015), and Selberg’s original papers.