View source for Zolotarev polynomials

''Zolotareff polynomials, Solotareff polynomials''

For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301201.png" />, the Zolotarev polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301202.png" /> is the unique solution of the problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301203.png" /></td> </tr></table>

That is, the Zolotarev polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301204.png" /> are the polynomials whose leading two coefficients are fixed, and that deviate least from zero on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301205.png" />, in the uniform norm (cf. also [[Uniform approximation|Uniform approximation]]; [[Polynomial least deviating from zero|Polynomial least deviating from zero]]).

This problem was formulated and solved by E.I. Zolotarev. It is the first of four Zolotarev problems in [[Approximation theory|approximation theory]]. Zolotarev's second problem asks for the monic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301206.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301207.png" /> that deviates least from zero on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301208.png" /> while satisfying the extra condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z1301209.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012011.png" />. The class of solutions (dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012013.png" />) are normalized Zolotarev polynomials. The third and fourth problems of Zolotarev deal with rational functions.

In each of these problems the influence of P.L. Chebyshev can be seen, whose lectures Zolotarev attended, as these problems are all generalizations of problems of Chebyshev. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012014.png" />, the monic Chebyshev polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012015.png" />, solves problem one but without any restriction on the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012016.png" />, cf. [[Chebyshev polynomials|Chebyshev polynomials]]. (From symmetry considerations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012017.png" />.)

It follows from the equi-oscillation theorem in the theory of best uniform approximation that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012018.png" /> is uniquely determined by the fact that the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012020.png" /> are as given, and it attains its norm, alternately in sign, at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012021.png" /> times in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012022.png" />. Thus it is easily verified that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012023.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012024.png" /></td> </tr></table>

(which is what one gets when the Chebyshev polynomial is fixed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012025.png" /> and pulled to the right as long as the resulting polynomial attains its norm alternately at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012026.png" /> points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012027.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012028.png" /> the Zolotarev polynomial is given in terms of elliptic integrals, see e.g. [[#References|[a1]]], [[#References|[a4]]], [[#References|[a6]]], [[#References|[a3]]] for more details. It suffices to consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012029.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012030.png" />.

Zolotarev polynomials, suitably normalized, are also the class of extremal polynomials in the pointwise Markov inequalities, cf. [[Markov inequality|Markov inequality]]. That is, they maximize

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012031.png" /></td> </tr></table>

for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012033.png" />, over all polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012034.png" /> which are bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012036.png" />. (For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012037.png" /> the extremal polynomial is always the Chebyshev polynomial.) This problem is considered in detail in [[#References|[a9]]] (see especially the index).

Various generalizations of Zolotarev polynomials can be found in the literature. The problem of finding the extremal polynomials when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012038.png" /> coefficients, rather than just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012039.png" /> coefficients, are fixed was considered by A. Akhiezer and by N.N. Meiman. For a generalization of the first Zolotarev problem to different (symmetric) domains and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012040.png" />, see [[#References|[a7]]] and references therein. Zolotarev perfect splines and Zolotarev <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012042.png" />-polynomials were introduced when generalizing the pointwise Markov inequalities to other spaces of functions, see e.g. [[#References|[a5]]], [[#References|[a8]]], and [[#References|[a2]]].

Due to fashions in transliteration, Zolotarev's name is sometimes written as Zolotareff or Solotareff.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Achieser,   "Theory of approximation" , F. Ungar  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.K. Bagdasarov,   "Zolotarev <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012043.png" />-polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012044.png" />"  ''J. Approx. Th.'' , '''90'''  (1997)  pp. 340–378</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.C. Carlson,   J. Todd,   "Zolotarev's first problem: the best approximation by polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130120/z13012047.png" />"  ''Aequat. Math.'' , '''26'''  (1983)  pp. 1–33</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Erdős,   G. Szegő,   "On a problem of I. Schur"  ''Ann. Math.'' , '''43'''  (1942)  pp. 451–470  (Correction: 74 (1961), 628; Reprinted in G. Szegő, Collected Papers (R.A. Askey, ed.), Vol. 2 (1982), 807-827, Birkhäuser)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Karlin,   "Oscillatory perfect splines and related extremal problems"  S. Karlin (ed.)  C.A. Micchelli (ed.)  A. Pinkus (ed.)  I.J. Schoenberg (ed.) , ''Spline Functions and Approximation Theory'' , Acad. Press  (1976)  pp. 371–460</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Paszkowski,   "The theory of uniform approximation I. Non-asymptotic theoretical problems"  ''Rozp. Mat.'' , '''26'''  (1962)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F. Peherstorfer,   K. Schiefermayr,   "Explicit generalized Zolotarev polynomials with complex coefficients II"  ''East J. Approx.'' , '''3'''  (1997)  pp. 473–483</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Pinkus,   "Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval"  ''J. Approx. Th.'' , '''23'''  (1978)  pp. 37–64</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.V. Voronovskaja,   "The functional method and its applications"  ''Transl. Math. Monographs'' , '''28'''  (1970)</TD></TR></table>