Conjectures of Bernstein and Erd\H os for weighted Lagrange interpolation on the halfline with exponential weights

Abstract: Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0<x_1<...<x_n=b}. Then the norm of the Lagrange interpolation operator is the maximum of the Lebesgue function L(x,t) on I. Bernstein conjectured that the norm of the Lagrange interpolation operator becomes minimal exactly for node systems which exhibit an equioscillation property in that the interval maxima m_k(x):=max_{[x_{k-1},x_k]} L(x,.)}, (k=1,...,n) are all equal. Erdős added to the conjecture the sandwich property that if y is an extremal (minimal norm) system, then for any other node system x there have to be indices i,j with m_i(y)<m_i(x) and m_j(y)> m_j(x). The conjectures were proved by Kilgore and de Boor--Pinkus in 1978. Since then, analogous results were obtained only for a few cases when interpolation is made to certain very special spaces of polynomials, or when we apply weighted interpolation with rather special weights. Worse than that, it turned out that published proofs of results on infinite intervals and weighted interpolation were seriously flawed. Here we prove the Bernstein and Erd\H os Conjectures for the case of exponentially weighted polynomials on the halfline. This is the first proof of these conjectures in a situation where, contrary to all existing successful proofs, we encounter singularity of certain derivative matrices.

• The paper establishes that a unique, equioscillating node system minimizes the operator norm for weighted Lagrange interpolation on [0,∞) using exponential decay.
• It introduces a finite-dimensional minimax reduction and employs extended Chebyshev-Haar systems to address singularities at interpolation endpoints.
• The study characterizes regular versus irregular node configurations, providing new tools for tackling approximation in unbounded, exponentially weighted domains.

Weighted Lagrange Interpolation with Exponential Weights and Conjectures of Bernstein and Erdős

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Introduction and Context

This paper addresses weighted Lagrange interpolation on the halfline $[0,\infty)$ using exponential weights, specifically $w(t)=\exp(-t)$. The focus lies on two classical conjectures regarding the minimax properties of the interpolation operator norm, attributed to Bernstein and Erdős. Although their conjectures were resolved in the unweighted, finite-interval case in the 1970s, the transition to weighted setups, particularly with exponential weights on the halfline, has encountered notable technical obstacles. Importantly, previous attempts to generalize the standard arguments—relying on properties like nonsingularity of derivative matrices—fail in this setting due to new types of singularity, especially at endpoints of the interpolation interval.

This work establishes, for the first time, the validity of the Bernstein and Erdős conjectures for Lagrange interpolation of real polynomials of bounded degree on $[0,\infty)$, weighted by $w(t)=\exp(-t)$, providing a comprehensive theoretical framework, new technical tools, and structural results indispensable for further generalizations.

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Theoretical Framework

The setting considers degree-$n$ Lagrange interpolation at nodes $0 = x_0 < x_1 < \cdots < x_n$ with respect to the weighted maximum norm:

$\|f\|_w = \sup_{t \ge 0} |f(t) w(t)|.$

The associated interpolation operator's norm is determined by the maximum of the weighted Lebesgue function $L_n(w, x, t)$ over $t \geq 0$, where

$L_n(w, x, t) = \sum_{k=0}^n |h_k(x, t)|$

and $w(t)=\exp(-t)$0 are weighted Lagrange basis functions.

Bernstein's Conjecture (Weighted Version)

The conjecture states that the minimal possible norm of the Lagrange interpolation operator occurs precisely when the maxima of the Lebesgue function on the $w(t)=\exp(-t)$1 subintervals between nodes are all equal ("equioscillation").

Erdős's Sandwich Conjecture

If $w(t)=\exp(-t)$2 is an extremal node system, then for any alternative node system $w(t)=\exp(-t)$3 there exist indices $w(t)=\exp(-t)$4, $w(t)=\exp(-t)$5 such that $w(t)=\exp(-t)$6 and $w(t)=\exp(-t)$7, where $w(t)=\exp(-t)$8 is the maximum of the Lebesgue function on the $w(t)=\exp(-t)$9th interval induced by $[0,\infty)$0.

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Methodological Advances and Main Results

1. Operator Norm Reduction to Finite-Dimensional Minimax

The core technical reduction is expressing the operator norm as a minimax of interval-wise maxima of the Lebesgue function, leading to a finite-dimensional optimization problem over the choice of interpolation nodes.

2. Analysis of Extended Chebyshev-Haar Systems (ECHS)

The analysis heavily employs the theory of ECHS, which enables strong statements about the uniqueness and localization of zeros and maxima of weighted polynomials and their derivatives.

3. Interlacing Properties

A central role is played by the interlacing of zeros of the weighted polynomials $[0,\infty)$1 and their derivatives, extending classical Markov-type results to exponentially weighted settings. The paper leverages Milev and Naidenov's results [MN, JMAA 2010] for exponential weights, ensuring these interlacing properties hold and enabling the structure theory for maxima and critical points of $[0,\infty)$2.

4. Breakdown and Circumvention of the "Standard Method"

The standard method—reliant on the nonsingularity of certain Jacobian matrices—breaks down in this context because local maxima of $[0,\infty)$3 may occur at the interval endpoints, leading to singular derivative matrices. The authors circumvent this by:

• Establishing precise regularity and differentiability properties of the maxima of the Lebesgue function as functions of the nodes, including the case where maxima lie at endpoints.
• Examining the structure of "regular" versus "irregular" node configurations, and characterizing the "fatness" and closure properties of the irregular set.

5. Properness and Homeomorphism of the Difference Mapping

Key to the global analysis is the properness of the difference mapping $[0,\infty)$4 and proof of its homeomorphism property (generalizing a main result of De Boor and Pinkus). This underwrites both existence and uniqueness of the extremal (equioscillating) node system.

6. Resolution of the Conjectures

Main Theorem: For exponentially weighted Lagrange interpolation on $[0,\infty)$5, there exists a unique node system achieving the minimal value of operator norm. It is characterized by equioscillation of the Lebesgue function's maxima over the subintervals.

Strong Numerical Statement: The Lebesgue constant is minimized uniquely at the equioscillating node system, and any deviation from this system leads to an increase in the maximum of the Lebesgue function.

Sandwich Property (Corollary): For any other node system $[0,\infty)$6, the extremal value for $[0,\infty)$7 is "sandwiched" between the minimum and maximum of $[0,\infty)$8, that is:

$[0,\infty)$9

Limitation of Generalized Majorization: Intriguingly, the sandwich property does not extend to a full-fledged intertwining property as in the classical case—the presence of singularities and the possible locking of interval maxima at the endpoints introduces fundamentally new phenomena.

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Implications and Future Directions

Theoretical Significance: The result extends foundational interpolation theory into genuinely new territory, demonstrating that minimax and equioscillation characterizations survive in infinite, weighted real domains with exponential decay. It also signals, via detailed counterexamples and technical obstacles, both the necessity to re-express some classical machinery and the subtleties that arise in such generalizations.

Technical Innovation: The developed methodology for handling singular points and the analysis of the regular/irregular node system dichotomy provide essential tools for any future extension of interpolation problems in weighted or infinite-dimensional settings—especially where traditional nonsingularity-based arguments fail.

Applications: While the application context is mainly theoretical approximation, exponential weights are central in orthogonal polynomial systems, quadrature, signal processing, and numerical schemes for unbounded domains, making this a fundamental advance.

Future Prospects: The authors identify several open avenues:

• Extension to more general weights and weighted function spaces.
• Further exploration of singularity-induced phenomena and their resolution.
• Potentials for multivariate or non-uniform settings, or incomplete/interpolatory polynomial spaces.

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Conclusion

The paper provides a rigorous, comprehensive solution to the Bernstein and Erdős conjectures for the weighted Lagrange interpolation operator norm in the challenging setting of exponential weights on the halfline. It overcomes longstanding obstacles by introducing technical refinements beyond the classical framework, proves the existence and uniqueness of equioscillating extremal node systems, determines sharp structural local and global properties, and elucidates the boundaries of classical majorization in this noncompact, weighted context. The methods and structural insights developed will serve as critical tools for subsequent research in weighted approximation theory and related areas.