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 $\{P_n(x)\}$ on the interval $[-1,1]$ with weight

$$\begin{array}{}\text{(1)}& h(x)=\frac{h_0(x)}{\sqrt{1-x^2}},x\in (-1,1),\end{array}$$

the question arises on the conditions of boundedness of the sequence $\{P_n(x)\}$ at a specific point, on a certain set $A\subset [-1,1]$ or on the whole interval of orthogonality. This question is important because when the sequence $\{P_n(x)\}$ 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

$$\begin{array}{}\text{(2)}& |P_n(x)|\le C_1,x\in A\subset [-1,1],\end{array}$$

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

$$\begin{array}{}\text{(3)}& h_0(x)\ge C_2>0,x\in A\subset [-1,1],\end{array}$$

be fulfilled. The value of the function $h_0$ at a point $x$ where the inequalities $\text{(2)}$ and $\text{(3)}$ are examined must be connected to the values of this function at the points close to $x$, and the problem consists of deducing $\text{(2)}$ from $\text{(3)}$, given minimal restrictions on the function $h_0$ in a neighbourhood of $x$ (the first Steklov problem). There are different local and global conditions (see [2], [5]) under which $\text{(2)}$ follows from $\text{(3)}$. In particular, if in $\text{(1)}$ the function $h_0$ is positive, continuous and satisfies certain extra conditions, then an asymptotic formula from which inequality $\text{(2)}$ for the polynomials $\{P_n(x)\}$ follows when $A=[-1,1]$ 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 [2]), if inequality $\text{(3)}$ is fulfilled on the entire interval $[-1,1]$, then there is a sequence $\{{ϵ}_n\}$, ${ϵ}_n>0$, ${ϵ}_n\to 0$, such that the inequality

$$|P_n(x)|\le {ϵ}_n\sqrt{n},x\in [-1,1],$$

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, 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 $(-1,1)$ 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).

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See Orthogonal polynomials for further details.