View source for Steklov problems

''in the theory of orthogonal polynomials''

Problems in which the asymptotic properties of orthogonal polynomials are studied in dependence on the properties and, particularly, on the singularities, of the weight function and the domain of orthogonality.

In the study of the [[Orthogonal polynomials|orthogonal polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877001.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877002.png" /> with weight

<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/s/s087/s087700/s0877003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

the question arises on the conditions of boundedness of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877004.png" /> at a specific point, on a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877005.png" /> or on the whole interval of orthogonality. This question is important because when the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877006.png" /> is bounded, certain properties of trigonometric Fourier series can be transferred to Fourier series in these orthogonal polynomials.

V.A. Steklov

proposed that for the inequality

<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/s/s087/s087700/s0877007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

to be fulfilled, it is necessary and sufficient that the condition

<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/s/s087/s087700/s0877008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

be fulfilled. The value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s0877009.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770010.png" /> where the inequalities (2) and (3) are examined must be connected to the values of this function at the points close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770011.png" />, and the problem consists of deducing (2) from (3), given minimal restrictions on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770012.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770013.png" /> (the first Steklov problem). There are different local and global conditions (see [[#References|[2]]], [[#References|[5]]]) under which (2) follows from (3). In particular, if in (1) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770014.png" /> is positive, continuous and satisfies certain extra conditions, then an asymptotic formula from which inequality (2) for the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770015.png" /> follows when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770016.png" /> holds.

Moreover, Steklov

examined cases of algebraic zeros of the weight function and established a series of results that served as the starting point of two directions of research. One of these is characterized by the so-called global, or uniform, estimation of the growth of orthonormal polynomials which are obtained under fairly general conditions on the weight function (the second Steklov problem). For example (see [[#References|[2]]]), if inequality (3) is fulfilled on the entire interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770017.png" />, then there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770020.png" />, such that the inequality

<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/s/s087/s087700/s08770021.png" /></td> </tr></table>

holds.

The third Steklov problem consists of studying the asymptotic properties of orthogonal polynomials given smooth singularities of the weight function. This course of research can also cover the asymptotic properties of the [[Jacobi polynomials|Jacobi polynomials]], the weight function of which has singularities at the end-points of the interval of orthogonality, hence the difference between the asymptotic properties of Jacobi polynomials within the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087700/s08770022.png" /> and at its end-points. The difference between results in the latter direction from global estimates of orthogonal polynomials is explained by the fact that in this case the weight function may, at certain points, vanish or become infinite of a definite order, and also from the fact that it satisfies certain conditions of smoothness. Asymptotic formulas and estimates for orthogonal polynomials are established separately at singular points of the weight function (zeros, poles, end-points of the interval of orthogonality) and on the rest of the interval of orthogonality.

The formulations and, especially, the proofs of all the above questions are most natural when the polynomials are orthogonal on the circle, as many results of the approximation of periodic functions by trigonometric polynomials can then be used (cf. also [[Orthogonal polynomials on a complex domain|Orthogonal polynomials on a complex domain]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> V.A. Steklov, "Une contribution nouvelle au problème de développement des fonctions arbitraires en série de polynômes de Tchebychef" ''Izv. Ross. Akad. Nauk.'' , '''15''' (1921) pp. 267–280</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> V.A. Steklov, "Une méthode de la solution du problème de développement des fonctions en séries de polynômes de Tchebychef indépendante de la théorie de fermeture I" ''Izv. Ross. Akad. Nauk.'' , '''15''' (1921) pp. 281–302</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> V.A. Steklov, "Une méthode de la solution du problème de développement des fonctions en séries de polynômes de Tchebychef indépendante de la théorie de fermeture II" ''Izv. Ross. Akad. Nauk.'' , '''15''' (1921) pp. 303–326</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Ya.L. Geronimus, "Polynomials orthogonal on a circle and interval" , Pergamon (1960) (Translated from Russian) {{MR|0133642}} {{ZBL|0093.26503}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) {{MR|0372517}} {{ZBL|0305.42011}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.K. Suetin, "Fundamental properties of polynomials orthogonal on a contour" ''Russian Math. Surveys'' , '''21''' : 2 (1966) pp. 35–83 ''Uspekhi Mat. Nauk'' , '''21''' : 2 (1966) pp. 41–88 {{MR|0198111}} {{ZBL|0182.09302}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.K. Suetin, "V.A. Steklov's problem in the theory of orthogonal polynomials" ''J. Soviet Math.'' , '''12''' : 6 (1979) pp. 631–681 ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''15''' (1977) pp. 5–82 {{MR|0493142}} {{ZBL|0473.42016}} </TD></TR></table>



====Comments====
See [[Orthogonal polynomials|Orthogonal polynomials]] for further details.