Abstract
The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).
Highlights
Denote by Pn1 the set of algebraic polynomials of degree n with the unit leading coefficient: p(z) = zn + cn−1zn−1 + . . . + c0, ck ∈ C.Consider the ellipse E = {z = a cos t + ib sin t | t ∈ [0, 2π]; a > b > 0} centered at the origin
Smirnov and Lebedev considered the ellipse as an image of a circle under the Joukowsky transform f (ω) = (ω + 1/ω) /2
Instead of polynomials, they studied functions of the following form defined on circles: qn(ω) = Rn(ω) + ω2 − 1 · Qn−1(ω), where Rn(ω) and Qn−1(ω) are polynomials of degree at most n and n − 1, respectively
Summary
Smirnov and Lebedev considered the ellipse as an image of a circle under the Joukowsky transform f (ω) = (ω + 1/ω) /2. Instead of polynomials, they studied functions of the following form defined on circles: qn(ω) = Rn(ω) + ω2 − 1 · Qn−1(ω), where Rn(ω) and Qn−1(ω) are polynomials of degree at most n and n − 1, respectively. We will give another solution of this problem. Let. be an ellipse with foci ±1, and let Pn[−1,1] be the set of algebraic polynomials of degree n with real coefficients and the unit uniform norm on [−1, 1]. Our idea and Kemperman’s one are similar, but our proof is shorter and easier to understand, as we consider objects in the problem from a slightly different point of view
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