[Paper Review] A Generalization of a Minimal Problem of Korkin-Zolotarev kind

Н. И. Ахиезер|arXiv (Cornell University)|Jan 22, 2014

Mathematical functions and polynomials1 references3 citations

TL;DR

This paper generalizes the Korkine-Zolotarev minimal problem to a weight function that is 1 on two disjoint intervals $[-1,\alpha]$ and $[\beta,1]$, and 0 on $(\alpha,\beta)$, solving for the monic polynomial of degree $n$ minimizing the $L^1$-norm under this weight. The solution is expressed exactly using elliptic functions for any $n$, $\alpha$, and $\beta$, and the structure of the extremal polynomial is characterized by interlacing zeros and a functional equation involving products of linear factors, with the key result being an exact closed-form solution via elliptic functions where previous methods only gave asymptotics.

ABSTRACT

This is an English translation of the paper in which N. I. Akhiezer discovered his famous orthogonal polynomials on two intervals in a connection with a generalization of the Korkin-Zolotarev (Korkine-Zolotaref) problem (see the small commentary by P. Yuditskii, attached to the current publication). Translated from German by F. Puchhammer.

Motivation & Objective

• Address the gap in the literature regarding minimal $L^1$-norm problems for monic polynomials under non-uniform, discontinuous weight functions.
• Extend the classical Korkine-Zolotarev problem—originally solved for $p(x) = 1$—to a weight function that vanishes on an interior subinterval $[\alpha, \beta]$.
• Characterize the extremal monic polynomial $f_n(x)$ of degree $n$ that minimizes $\int_E |f_n(x)| \, dx$, where $E = [-1,\alpha] \cup [\beta,1]$.
• Establish that the solution is expressible in terms of elliptic functions for any $n$, $\alpha$, and $\beta$, overcoming limitations of prior asymptotic methods.
• Prove that the extremal polynomial has simple real zeros confined to $[-1,\alpha]$ and $[\beta,1]$, with no zeros in $(\alpha,\beta)$ except under specific integral balance conditions.

Proposed method

• Formulate the minimization problem as finding the monic polynomial $f_n(x) = x^n + p_1x^{n-1} + \cdots + p_n$ that minimizes $I[f] = \int_{-1}^{\alpha} |f(x)| \, dx + \int_{\beta}^{1} |f(x)| \, dx$.
• Derive the critical point conditions $\partial I[f]/\partial p_i = 0$ for $i = 0,1,\dots,n-1$, leading to a system of integral equations involving powers of $x$.
• Use the method of Tchebyshev and Korkine-Zolotarev to transform the system into a functional equation involving the Stieltjes transform of the sign variation of $f(x)$.
• Introduce the zero structure of $f_n(x)$: all zeros $\xi_1 < \cdots < \xi_n$ lie in $[-1,\alpha] \cup [\beta,1]$, with no zeros in $(\alpha,\beta)$ unless a balance condition on the $L^1$-norms holds.
• Define even and odd index products $U_m(x)$ and $V_{m+1}(x)$ of the zeros to derive algebraic identities: for odd $n=2m+1$, $(x+1)(x-\beta)(x-1)U_m^2(x) - (x-\alpha)V_{m+1}^2(x) = Ax + B$, and for even $n=2m$, a similar identity with $\alpha$ and $\beta$ swapped.
• Show that the resulting algebraic equation defines a hyperelliptic curve, whose solution is expressible in terms of elliptic functions, yielding an exact closed-form solution for all $n$, $\alpha$, and $\beta$.

Experimental results

Research questions

• RQ1 What is the exact form of the monic polynomial of degree $n$ that minimizes the $L^1$-norm under a two-interval weight function $p(x) = 1$ on $[-1,\alpha] \cup [\beta,1]$ and $0$ on $(\alpha,\beta)$?
• RQ2 How does the extremal polynomial's zero distribution behave, and under what conditions can it have a zero in $(\alpha,\beta)$?
• RQ3 What is the role of the balance condition $\int_{-1}^{\alpha} |\varphi(x)| \, dx = \int_{\beta}^{1} |\varphi(x)| \, dx$ in determining the existence of interior zeros?
• RQ4 Can the minimal $L^1$-norm problem for such a discontinuous weight be solved exactly using special functions, and if so, which ones?
• RQ5 Does the solution for the two-interval case yield a closed-form expression in terms of elliptic functions for all $n$, $\alpha$, and $\beta$, unlike previous asymptotic approaches?

Key findings

• The extremal monic polynomial minimizing $\int_E |f_n(x)| \, dx$ has all its $n$ zeros simple and located in $[-1,\alpha] \cup [\beta,1]$, with no zeros in $(\alpha,\beta)$ unless a specific integral balance condition holds.
• An interior zero $\xi \in (\alpha,\beta)$ is only possible if $\int_{-1}^{\alpha} \frac{|f(x)|}{\xi - x} \, dx = \int_{\beta}^{1} \frac{|f(x)|}{x - \xi} \, dx$, and in such cases, the solution is not unique.
• For $n = 2m+1$ odd, the extremal polynomial satisfies the algebraic equation $(x+1)(x-\beta)(x-1)U_m^2(x) - (x-\alpha)V_{m+1}^2(x) = Ax + B$, where $U_m$ and $V_{m+1}$ are products of even- and odd-indexed zeros.
• For $n = 2m$ even, the corresponding equation is $(x+1)(x-\alpha)(x-1)U_m^2(x) - (x-\beta)V_{m+1}^2(x) = Ax + B$, with the same structure but with $\alpha$ and $\beta$ swapped in the coefficients.
• The solution to the minimization problem is expressible in terms of elliptic functions for any $n$, $\alpha$, and $\beta$, providing an exact closed-form solution, a significant improvement over prior asymptotic results.
• The problem's solution is unique up to the choice of zero placement in $[\alpha,\beta]$ when such zeros exist, but these can be replaced by zeros at $\alpha$ or $\beta$ without changing the value of $I[f]$, allowing the restriction to zeros strictly in $[-1,\alpha] \cup [\beta,1]$.