ON ONE PROBLEM OF C. DAVIS

Verdi G. Verdiev

In this paper we present a rather general method of the proof of the existence of functions with prescribed extreme values. The method is based on the theory of extremal problems.

1. Introduction

In 1956 C. Davis ([1], [2]) posed the following problem: For a real polynomial p(·) of degree N, denote the zeros of p′(·) (multiplicity counted) by τ_{i}, i = 1, …, N − 1. Assume that τ_{i} are real, and let τ_{1} ≤ ⋯ ≤ τ_{N−1}. Now to what extent are the numbers p(τ_{i}) arbitrary? More precisely, give necessary and sufficient conditions on an (N − 1)-tuple of real numbers v_{1}, …, v_{N−1} in order that there exists a polynomial p(·) such that p′(τ_{i}) = 0, p(τ_{i}) = v_{i}, i = 1, …, N − 1.

C. Davis solved this problem in 1957 ([2]) for algebraic polynomials. Later many investigators (see [3]–[8]) gave other solutions of this problem and solutions of an analogous setting for Chebyshev systems of polynomials and splines. It should be pointed out that in [6], [7] more general results are obtained.

These results are formulated below (see Theorem 1.1 and 1.2).

2. Theorems on existence of an algebraic polynomial and a spline with prescribed extreme values

Let Δ:= [0, 1], m, N, n ∈ ℕ, T:= {t_{i}}_{i=1}^{n} ⊂ Δ, and 0 ≤ t_{1} < ⋯ < t_{n} ≤ 1. Denote by P_{N} the family of all algebraic polynomials of order N and by S_{m}^{n} the family of m-order polynomial splines of defect 1 on Δ with nodes at the points t_{i} ∈ T.

Definition. A sequence

V = {v_{i}}_{i=0}^{N}, v_{i} ∈ ℝ, i = 0, 1, …, N,

satisfying the condition

(1.1)  (v_{i+1} − v_{i}) (v_{i} − v_{i−1}) < 0, i = 1, …, N − 1,

is called a sequence of extreme values (s. e. v.).

Theorem 1.1. Let V = {v_{i}}_{i=0}^{N} be a s. e. v. Then there exists a unique algebraic polynomial p̂(·) ∈ P and a unique system of points {τ̂_{i}}_{i=0}^{N} ⊂ Δ,

0 = τ̂_{0} < τ̂_{1} < ⋯ < τ̂_{N} = 1,

such that

(1.2)  p̂(τ̂_{i}) = v_{i}, i = 0, 1, …, N,
(1.3)  p̂′(τ̂_{i}) = 0, j = 1, …, N − 1.

Theorem 1.2. Let m, n, N ∈ ℕ, m + n = N, and let V = {v_{i}}_{i=0}^{N} be a s. e. v. Then there exists a unique spline ŝ(·) ∈ S_{m}^{n} and a unique system of points

{θ̂_{i}}_{i=0}^{N} ⊂ Δ, 0 = θ̂_{0} < θ̂_{1} < ⋯ < θ̂_{N} = 1,

such that

\begin{aligned}ŝ(θ̂_{i}) & = v_{i}, i = 0, 1, …, N, & (1.4) \\ s′(θ̂_{j}) & = 0, j = 1, …, N − 1, & (1.5) \end{aligned}

θ̂_{i} < t_{i} < θ̂_{i+m}, i = 1, …, n. (1.6)

3. Setting of extremal problems

Denote by

Σ := {μ = (μ_{1}, …, μ_{N−1}): μ_{i} ≥ 0, ∑_{i=1}^{N−1} μ_{i} = 1}

the (N − 1)-dimensional simplex. Put

Σ_{T} := {η = (η_{1}, …, η_{N−1}) ∈ Σ: ∑_{j=1}^{i} η_{j} < t_{i} < ∑_{j=1}^{m+i} η_{j}, t_{i} ∈ T }.

Let us introduce the following notation:

τ_{0}(μ) := 0, τ_{i}(μ) := ∑_{j=1}^{i} μ_{j}, θ_{0}(η) := 0, θ_{i}(η) := ∑_{j=1}^{i} η_{j}, i = 1, 2, …, N.

Given any s. e. v.

V = {v_{i}}_{i=0}^{N},

we construct a class of algebraic polynomials P and a class of splines S as follows

P := {p(·) ∈ P_{N}: p(τ_{i}(μ)) = v_{i}, μ ∈ int Σ, v_{i} ∈ V },
S := {s(·) ∈ S_{m}^{n}: s(θ_{i}(η)) = v_{i}, η ∈ int Σ_{T} , v_{i} ∈ V }.

Let us consider the value p(2) (p(·) ∈ P) as a function of a point μ = (μ_{1}, …, μ_{N−1}) ∈ int Σ, that is, p(2) := p(2, μ), and the value s(2) (s(·) ∈ S) as a function of a point η = (η_{1}, …, η_{N−1}) ∈ int Σ_{T}, that is, s(2) := s(2, η). Now we can formulate the following extremal problems.

Problem 3.1.

p(2) := p(2, μ) → inf, p(·) ∈ P, μ ∈ int Σ, if v_{N−1} < v_{N},

and

p(2) := p(2, μ) → sup, p(·) ∈ P, μ ∈ int Σ, if v_{N−1} > v_{N}.

Problem 3.2.

s(2) := s(2, η) → inf, s(·) ∈ S, η ∈ int Σ_{T} , if v_{N−1} < v_{N},

and

s(2) := s(2, η) → sup, s(·) ∈ S, μ ∈ int Σ_{T} , if v_{N−1} > v_{N}.

4. Solution of extremal problems 3.1 and 3.2

Theorem 4.1. There exists

μ̂ = (μ̂_{1}, …, μ̂_{N−1}) ∈ int Σ

such that

(4.1)  inf_{p(·)∈P} p(2) = inf_{μ∈Σ} p(2, μ) = p(2, μ̂) if v_{N−1} < v_{N},

and

(4.2)  sup_{p(·)∈P} p(2) = sup_{μ∈Σ} p(2, μ) = p(2, μ̂) if v_{n−1} > v_{n}.

Theorem 4.2. There exists η̂ = (η̂_{1}, …, η̂_{n−1}) ∈ int Σ_{T} such that

(4.3)  inf_{s(·)∈S} s(2) = inf_{η∈Σ_{T}} s(2, η) = s(2, η̂) if v_{n−1} < v_{n},

and

(4.4)  sup_{s(·)∈S} s(2) = sup_{η∈Σ_{T}} s(2, η) = s(2, μ̂) if v_{n−1} > v_{n}.

The idea of proofs of Theorems 4.1 and 4.2 is based on the Weierstrass compactness principle. Since p(2) = p(2, μ) (p(·) ∈ P) and s(2) = s(2, η) (s(·) ∈ S) are continuous functions on Σ and Σ_{T}, respectively, and

lim_{int Σ∋μ→μ^{∘}∈∂Σ} p(2, μ) = \{ \begin{matrix}+∞, & if v_{n−1} < v_{n} \\ −∞, & if v_{n−1} > v_{n}, \end{matrix}

and

lim_{int Σ_{T} ∋η→η^{∘}∈∂Σ_{T}} s(2, η) = \{ \begin{matrix}+∞, & if v_{n−1} < v_{n} \\ −∞, & if v_{n−1} > v_{n}, \end{matrix}

then by the Weierstrass theorem there exist points μ̂ ∈ int Σ and η̂ ∈ int Σ_{T} such that identities (4.3) and (4.4) are valid.

5. The scheme of proofs of Theorems 1.1 and 1.2

Beginning with the points μ̂ = (μ̂_{1}, …, μ̂_{n−1}) ∈ int Σ and η̂ = (η̂_{1}, …, η̂_{n−1}) ∈ int Σ_{T} (which were found in Theorems 4.1 and 4.2), we construct the points

τ̂ = (τ̂_{0}, τ̂_{1}, …, τ̂_{n}) ⊂ Δ and θ̂ = (θ̂_{0}, θ̂_{1}, …, θ̂_{n}) ⊂ Δ

so that

τ̂_{0} := 0, τ̂_{i} = ∑_{j=1}^{i} μ_{i}, θ̂_{0} := 0, θ̂_{i} := ∑_{j=1}^{i} η̂_{i}, i = 1, …, N.

Then, we find an interpolation polynomial p̂(·) ∈ R_{N} and an interpolation spline ŝ(·) ∈ S_{m}^{n} satisfying the conditions:

p̂(τ̂_{i}) = v_{i}, i = 0, 1, …, N;
ŝ(θ̂_{i}) = v_{i}, i = 0, 1, …, N.

The polynomial p̂(·) ∈ P_{N} and the spline ŝ(·) ∈ S_{m}^{n} are uniquely defined and they satisfy the assumptions of Theorems 1.1 and 1.2.

6. Generalizations

The represented method enables to solve various generalizations of the Davis problem in many directions.

References

[1] C. Davis, Problem 4714, Amer. Math. Monthly 63 (1956), no. 10, 729.

[2] C. Davis, Extrema of a polynomial, Amer. Math. Monthly 64 (1957), no. 9, 679–680.

[3] J. Mucielski and S. Paszkowski, A generalization of Tschebysheff polynomials, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 8 (1960), no. 7, 433–438.

[4] W. J. Kammerer, Polynomial approximations of finitely oscillating functions, Math. Comp. 15 (1961), no. 74, 115–119.

[5] J. Kudela, Construction des certains polynômes extrémaux, Prace Matematyczne LXXVII (1963), no. 9, 31–36.

[6] V. S. Videnskii, The existence and uniqueness of a solution of an interpolation problem, Sb. Nauchn. Tr. Leningr. Mech. Inst. (1965), no. 50, 29–41.

[7] S. H. Fitzgerald and L. L. Shumaker, A differential equation approach to interpolation at extremal points, J. Analyse Math. 22 (1969), 117–139.

[8] M. B. Korobkova, A theorem on the existence of a spline-polynomial with a given sequence of extrema, Mat. Zametki 11 (1972), 251–258.