Editorial: S. N. Mergelyan’s Dissertation “Best Approximations in the Complex Domain” (Short Version)

1. Direct Theorems for Domains with Different Types of Singularities

Let us present the formulation of one of Warschawski’s results, which we will use in the future.

Let D be a Jordan domain bounded by a curve $\Gamma$ passing through $z=0$. Suppose that in the neighborhood $\left|z\right|\le a$ of the point $z=0$ the boundary $\Gamma$ of the domain D consists of two arcs ${\Gamma }_{+}$ and ${\Gamma }_{-}$, the equations of which in polar coordinates are

$$\phi ={\Phi }_{+}\left(\rho \right),\phi ={\Phi }_{-}\left(\rho \right)({\Phi }_{+}<{\Phi }_{-})$$

respectively. Let there also be limits

$$\underset{\rho \to 0}{lim}\rho \frac{d{\Phi }_{+}}{d\rho };\underset{\rho \to 0}{lim}\rho \frac{d{\Phi }_{-}}{d\rho }.$$

Let $w=w\left(z\right)$ denote the function that conformally maps the domain D onto the circle $|w-1|<1$ so that $z=0$ goes to $w=0$; $\theta \left(\rho \right)={\Phi }_{-}\left(\rho \right)-{\Phi }_{+}\left(\rho \right)$.

Theorem 1.1 

(S. E. Warschawski [5]). If

$$\underset{0}{\overset{a}{\int }}\left[{\left(\frac{d{\Phi }_{+}\left(\rho \right)}{d\rho }\right)}^2+{\left(\frac{d{\Phi }_{-}\left(\rho \right)}{d\rho }\right)}^2\right]\frac{\rho d\rho }{\theta \left(\rho \right)}<\infty ,$$

then in the neighborhood of the point $z=0$ we have

$$\left|w\left(z\right)\right|=Cexp\left\{-\pi \underset{\left|z\right|}{\overset{a}{\int }}\frac{d\rho }{\rho \theta \left(\rho \right)}+O\left(1\right)\right\},$$

(1.1)

where C does not depend on z.

This theorem makes it possible in a number of cases to investigate the behavior of conformal mapping functions in a closed domain, as well as to estimate the distance of the level lines of the Green’s function to a boundary point, depending on the behavior of the boundary near this point.

Let D be a bounded domain with a simply connected complement; henceforth, by ${\mathcal{Z}}_R$$(R>1)$ we mean the level line of the Green’s function $G\left(z\right)$ of the complement to $\overline{D}-{\mathcal{Z}}_R;G\left(z\right)=\ln R$.

$D_R$ is a bounded domain bounded by ${\mathcal{Z}}_R$. If $f\left(z\right)$ is regular in $D_R$ and does not exceed unity there in absolute value, then for any integer $n\ge 1$ there is a polynomial ${\mathcal{P}}_n\left(z\right)$ of degree n, for which

$$\underset{z\in \overline{D}}{max}|f\left(z\right)-{\mathcal{P}}_n\left(z\right)|<\frac{C_0}{{(R-1)}^3{R}^n},$$

(1.2)

where $C_0$ is an absolute constant (the degree of difference $R-1$ can be reduced, however the question of determining it as accurately as possible does not interest us now).

The proof is easy to derive by composing a Fejér interpolation polynomial with uniformly distributed nodes, estimating the remainder term represented by the Cauchy integral, and considering that, firstly, if the diameter D is less than one, then

$$\mathit{length}{\mathcal{Z}}_R<\frac{C_0^{\prime }}{R-1},$$

and secondly, the distance of any point ${\mathcal{Z}}_R$ to $\Gamma$ exceeds $\frac{1}{C_0^{\prime }}{(R-1)}^2$, where $C_0^{\prime }$ is an absolute constant.

Let D now contain $z=0$ and the equation of its boundary in polar coordinate be

$$\rho =\rho \left(\phi \right),$$

where $\rho \left(\phi \right)=\rho (\phi +2\pi )$ is a single-valued continuous function.

Theorem 1.2. 

If all derivative numbers of the function $\ln \rho \left(\phi \right)$ are uniformly bounded from above by the number k and the function $f\left(z\right)$ is regular in D, has a continuous $m^{th}$ derivative in $\overline{D}$, the modulus of continuity of which is $w\left(\delta \right)$, then

$${\rho }_n(D;f)<const{\left(\frac{\ln n}{n}\right)}^{\frac{2m}{\pi }arctg\frac{1}{k}}w\left({\left(\frac{\ln n}{n}\right)}^{\frac{2}{\pi }arctg\frac{1}{k}}\right).$$

(1.3)

Let D be bounded by a finite number of smooth curves that make angles with each other, the internal openings of which do not exceed $2\pi \alpha$, and $f^{\left(m\right)}\left(z\right)$ be regular in D and satisfy the Lipschitz condition of order $\gamma$ in $\overline{D}$.

Assuming the domain D is star-shaped with respect to one of its points, it is easy to show by the reasoning used to obtain estimate (1.3) that

$$\rho (D;f)<\frac{C\left(\epsilon \right)}{n^{2(1-\alpha )(m+\gamma )-\epsilon }},n=1,2,3,\dots ,$$

(1.4)

for any $\epsilon >0$.

We can free ourselves from the artificial restriction of star-shapedness by using an additional reasoning based on the “averaging” method of academician M.V. Keldysh [6], which we set out in §3 of Theorem 1.3.

If no additional restrictions are imposed on the smoothness of the boundary D, then, as will be seen, in §3, the numbers $C\left(\epsilon \right)$ can increase arbitrarily quickly: for any positive function $f\left(z\right)$ there exists $N\left(\epsilon \right)$ such that $f^{\left(m\right)}\left(z\right)$ satisfies a Lipschitz condition of order $\gamma$ in $\overline{D}$; however,

$$\rho (D;f)>\frac{N\left(\epsilon \right)}{n^{2(1-\alpha )(m+\gamma )-\epsilon }},n_1\left(\epsilon \right)<n<n_2\left(\epsilon \right).$$

Now, let the domain D have an incoming point, and $\rho =\rho \left(\phi \right)$ still means the equation of the boundary in polar coordinates; assume that ${\rho }^{\prime }\left(\phi \right)$ exists everywhere outside $\phi =0$, and also that at the point $(0;\rho (0\left)\right)$ two arcs of the boundary of D touch the axis $OX$, and ${\rho }^{\prime }\left(\phi \right)$ decreases monotonically as $\phi \to +0$, $\phi \to 2\pi -0$; by $d\left(\alpha \right)$ we denote the distance of the point $(0;\rho (0\left)\right)$ to the level line ${\mathcal{Z}}_{1+\alpha }$.

Theorem 1.3. 

If $f\left(z\right)$ is regular in D, its $m^{th}$ derivative has $w\left(\delta \right)$ modulus of continuity in $\overline{D}$, then

$${\rho }_n(D;f)<const{\left(d\left(\frac{\ln n}{n}\right)\right)}^{m}w\left(d\left(\frac{\ln n}{n}\right)\right).$$

(1.5)

This circumstance constitutes a distinguishing feature of the best approximations in the complex domain from the best approximations in the real domain.

2. Inverse Theorems

Let D denote a domain bounded by the Jordan curve $\Gamma$. By $d(\xi ;\alpha )$ we denote the distance of the image of the circle $\left|w\right|=1+\alpha$$(\alpha >0)$ under a conformal mapping of the exterior of the unit circle to the complement of $\overline{D}$ to a boundary point $\xi$.

Let $B_{\xi }$ denote an arbitrary subdomain of D having only the property that the ratio of the distance of any point of $B_{\xi }$ to $\xi$ to the distance of the same point to $\Gamma$ is bounded from above uniformly with respect to all points of $B_{\xi }$. The class of functions that are regular in some domain G and whose $k^{th}$ derivative satisfies a Lipschitz condition of order ${\alpha }^{\prime }\le 1$ in $\overline{G}$ is denoted by $Z(G;k+{\alpha }^{\prime })$ $({\alpha }^{\prime }>0)$.

It should be noted that by the modulus of continuity $w\left(\delta \right)$ of the function $f\left(z\right)$ in $\overline{G}$ we mean the supremum of the quantities

$$|f\left(z^{\prime }\right)-f\left(z^{″}\right)|$$

by all possible pairs $z^{\prime },z^{″}$ belonging to $\overline{G}$ and such that $z^{\prime }$ can be connected to $z^{″}$ by a rectifiable curve lying entirely in $\overline{G}$ and by length not exceeding $\delta$; for some domains this definition obviously does not coincide with the definition of the modulus of continuity that is given in the real domain; accordingly, a different meaning, generally speaking, is attached to the satisfaction of the Lipschitz condition in a closed domain $\overline{G}$.

It is easy to see that the quantity $w\left(\delta \right)$ is closely related to the properties of the function $f\left(z\right)$, while

$$w_1\left(\delta \right)=\underset{|z^{\prime }-z^{″}|\le \delta ,z^{\prime },z^{″}\in \overline{G}}{max}|f\left(z^{\prime }\right)-f\left(z^{″}\right)|$$

represents an artificial formation in relation to $f\left(z\right)$.

Indeed, for any function $\mu \left(\delta \right)$ decreasing monotonically to zero, one can construct a domain such that the fact that $w_1\left(\xi \right)<\mu \left(\xi \right)$ implies infinite differentiability of $f\left(z\right)$ at individual points on the boundary of the domain. As a similar example, we can take a domain with an incoming point $z=0$ and a sufficiently large order of contact of two boundary arcs at $z=0$.

The following proposition also applies to this question, the proof of which we will not dwell on.

Proposition 2.1. 

If two domains $D_1$ and $D_2$ have one common boundary point $z=0$, and the function $f\left(z\right)$ in $D_i$ coincides with the function $f_i\left(z\right)$ that is regular in $D_i$ and continuous in ${\overline{D}}_i$$(i=1,2)$, and $f_1\left(0\right)=f_2\left(0\right)$, then for any function $\nu \left(\delta \right)$ decreasing towards zero one can specify such a large order of contact of the boundaries of $D_1$ and $D_2$ at $z=0$ that from the inequality

$$w_1\left(\delta \right)<\nu \left(\delta \right)$$

(2.1)

it follows that if one of the functions $f_1\left(z\right),f_2\left(z\right)$ is identically equal to zero in the corresponding domain, then the same can be stated regarding the other function, i.e., pairs of functions satisfying (2.1) constitute a quasi-analytic class.

This Proposition can be deduced from a theorem to be proved later.

Theorem 2.1. 

If

$$\underset{n\to \infty }{lim}inf\frac{\ln \rho \left(n\right)}{\ln d(\xi ;\frac{1}{n})}=A,$$

(2.2)

then for any $\epsilon >0$, $f\left(z\right)\in \mathcal{Z}(B_{\xi };A-\epsilon )$; if $A=\infty$, $f\left(z\right)$ is infinitely differentiable in ${\overline{B}}_{\xi }$.

Corollary 2.1. 

From the above reasoning it can be seen that for $A=\infty$$f\left(z\right)$ is infinitely differentiable in ${\overline{B}}_{\xi }$.

Thus, an arbitrarily slow approximation rate $\rho \left(n\right)$ ensures infinite differentiability of the approximated function at some boundary points, if only the domain is located appropriately near these points.

Applying Warschawski’s result on conformal mapping stated above, in many cases it is possible, by estimating $d(\xi ;\alpha )$, to formulate the previous theorem directly in terms of the boundary of the domain.

Let the domain D contain, for some $\alpha >0$, the segment $(-\alpha ,0)$ and its boundary near $z=0$ be determined by the equation

$$\begin{array}{c}\left|y\right|=\phi \left(x\right),0<x<\alpha ,\\ C_1{x}^m<\phi \left(x\right)<C_2{x}^m,m>1,\end{array}$$

that is, we have a domain with an incoming point of algebraic tangent order.

In this case, according to Theorem 1.1,

$$d(0;\epsilon )>\frac{C}{{\left|\ln \epsilon \right|}^{\frac{1}{m-1}}};$$

(2.3)

therefore, it can be proved that

Corollary 2.2. 

If for some $C>0$

$$\rho \left(n\right)<\frac{C}{{\left(\ln n\right)}^p},p>0,$$

then $f\left(z\right)\in \mathcal{Z}(B_0;p(m-1)-\epsilon )$ for any $\epsilon >0$; if the proof is carried out carefully, then it can be shown that even $f\left(z\right)\in \mathcal{Z}(B_0;p(m-1))$, and this result is quite accurate, in the sense that, as it follows from Theorem 2.1, it is invertible: if $f\left(z\right)\in \mathcal{Z}(D;p(m-1\left)\right)$, then $\rho \left(n\right)<\frac{C}{{\left(\ln n\right)}^p}$.

Now, let $A=\infty$. We denote

$$M_n=\underset{z\in {\overline{B}}_{\xi }}{max}\left|f^{\left(n\right)}\left(z\right)\right|.$$

The rate of increase of the numbers $M_n$ depends on the rate of decrease of the numbers $\rho \left(n\right)$ and will be closer to the rate of increase of $C^{n}n!$, the closer $\rho \left(n\right)$ to any geometric progression. Namely, we will show that for any $\mu$, $0<\mu <1$

$$M_n<C^{n}n!\sum _{k=1}^{\infty }\frac{{\left(\rho \left(n_{k-1}\right)\right)}^{\mu }}{{\left(d\left(\xi ;\frac{1-\mu }{n_k}\ln \frac{1}{\rho \left(n_k\right)}\right)\right)}^n}.$$

(2.4)

In the case of $\rho \left(n\right)<q^n$$(q<1)$ this shows that $M_n<C_0^{n}n!$ ($C_0$ does not depend on n), i.e., $f\left(z\right)$ is analytic in $\overline{D}$.

Indeed, for any $\alpha >0$, using the Cauchy integral, one can obtain the estimate

$$|{\mathcal{P}}_{n_k}^m\left(z\right)-{\mathcal{P}}_{n_{k-1}}^m\left(z\right)|<C^{m}m!\frac{\rho \left(n_{k-1}\right){(1+\alpha )}^{n_k}}{{\left(d(\xi ;\alpha )\right)}^m};$$

(2.5)

putting $\alpha =\frac{1-\mu }{n_k}\ln \frac{1}{\rho \left(n_k\right)}$ and taking into account (2.5) we obtain, summing over k, the required inequality. In particular, let

$$d(\xi ;\alpha )\simeq {\alpha }^{\beta }(0<\beta <1)\mathrm{and}\rho \left(n\right)<e^{-n^{\alpha }}(0<\alpha <1).$$

In this case, it is easy to calculate the right-hand side of (2.4):

$$M_n<C^{n}n!n^{n\frac{\beta (1-\alpha )}{\alpha }}$$

(2.6)

For $\alpha$ close to one, the estimate (2.6) does not differ much from the exact one. Indeed, consider the function

$$\mathcal{F}\left(z\right)={\int }_0^1{e}^{-{\phi }^{-1}\left(d^{-1}\left(x\right)\right)}\sqrt{z-x}dx,$$

(2.7)

where $\phi \left(n\right)=\frac{1}{n}\ln \frac{1}{\rho \left(n\right)}$, and ${\phi }^{-1}$ and ${\rho }^{-1}$ are functions inverse to the corresponding ones, and z belongs to the domain D, symmetrically located relative to the $OX$ axis so that the distance of its boundary point $z=0$ to the level line ${\mathcal{Z}}_{1+\epsilon }$ is equal to $d(0;\epsilon )=d\left(\epsilon \right)$ and $\overline{D}$ does not intersect with $argz=0$ anywhere other than $z=0$.

Since the function

$${\mathcal{F}}_{\alpha }\left(z\right)={\int }_{d(0;\alpha )}^1{e}^{-{\phi }^{-1}\left(d^{-1}\left(x\right)\right)}\sqrt{z-x}dx$$

is regular in the domain $D_{1+\alpha }$ bounded by ${\mathcal{Z}}_{1+\alpha }$, then, as is known, there exists a polynomial ${\mathcal{P}}_n\left(z\right)$ so that

$$\underset{z\in \overline{D}}{max}|{\mathcal{F}}_{\alpha }\left(z\right)-{\mathcal{P}}_n\left(z\right)|<C\frac{1}{{\alpha }^2{(1+\alpha )}^n}(C=const).$$

But

$$\underset{z\in \overline{D}}{max}|\mathcal{F}\left(z\right)-{\mathcal{F}}_{\alpha }\left(z\right)|<{\int }_0^{d(0;\alpha )}{e}^{-{\phi }^{-1}(d^{-1}}\left(x\right)\sqrt{z-x}dx<e^{-{\phi }^{-1}\left(\alpha \right)}.$$

Setting $\alpha =\phi \left(\right[n\phi \left(n\right)\left]\right)$ we obtain

$$\underset{z\in \overline{D}}{max}|\mathcal{F}\left(z\right)-{\mathcal{P}}_n\left(z\right)|<C\rho \left(n\right).$$

Let us estimate $|{\mathcal{F}}^{\left(n\right)}\left(0\right)|$ from below. We have

$$|{\mathcal{F}}^{\left(n\right)}\left(0\right)|=\frac{1·3\cdots \overline{2n-1}}{2^n}\underset{0}{\overset{1}{\int }}{\left(\frac{1}{x}\right)}^{\frac{2n+1}{2}}{e}^{-{\phi }^{-1}\left(d^{-1}(0;x)\right)}dx.$$

Estimating the integral and assuming the existence of $d^{″}(0;\alpha )$, ${\phi }^{″}\left(x\right)$, we obtain

$$|{\mathcal{F}}^{\left(n\right)}\left(0\right)|\ge C^{n}n!\underset{1}{\overset{\infty }{\int }}\frac{d^{\prime }(0;\phi \left(x\right)){\phi }^{\prime }\left(x\right)e^{-x}dx}{{\left(d(0;\frac{1}{x}\ln \frac{1}{\rho \left(x\right)})\right)}^n}.$$

(2.8)

In our case $d(0;\alpha )\simeq {\alpha }^{\beta }$ and $\rho \left(x\right)\simeq e^{-x^{\alpha }}$; substituting in (2.8) we have

$$M_n\ge \left|{\mathcal{F}}^n\left(0\right)\right|>C^{n}n!n^{n\beta (1-\alpha )},$$

that is, for small $1-\alpha$ the estimate (2.6) does not differ much from the exact one.

So, for an arbitrarily slow rate of approximation, the corresponding structure of the domain can guarantee sufficiently “good” properties of the function at some boundary points. Therefore, the question arises: is it possible, by means of an appropriate construction of the domain, to ensure that the rate of approximation, different from the progression, would guarantee the analyticity of the function at some points on the boundary?

Regarding this question, the following can be stated:

Theorem 2.2. 

For any function $\phi \left(n\right)>0$ satisfying, for any $k>1$, the condition

$$\underset{n\to \infty }{lim}{k}^n\phi \left(n\right)=\infty$$

(2.9)

and any bounded domain D of the Carathéodory class, there exists a function $f\left(z\right)$ regular in D, continuous in $\overline{D}$, the rate of approximation to which satisfies the inequality

$$\rho \left(n\right)<\phi \left(n\right),$$

(2.10)

however for which the boundary of D is a cut.

Now consider the case when

$$A=\underset{n\to \infty }{lim}inf\frac{\ln {\rho }_n(f;D)}{\ln d(\xi ;\frac{1}{n})}=0.$$

In this case, $f\left(z\right)$ may already be non-differentiable and not satisfy any Lipschitz condition of positive order; however, its modulus of continuity $\omega \left(\delta \right)$ in ${\overline{B}}_{\xi }$ satisfies the following constraint.

Theorem 2.3. 

For some $C>0$ independent of δ

$$\omega \left(\delta \right)<C\underset{n\ge 1}{min}\left\{{\rho }_n(f;D)+\frac{\delta }{d(\xi ;\frac{1}{n})}\right\}.$$

It will be shown later that, in the general case, this estimate cannot be improved.

Let us note one corollary of Theorem 2.1.

Corollary 2.3. 

Let the domain D be convex or bounded by a polygonal line with a finite number of segments. If $f\left(z\right)$ can be approached in $\overline{D}$ with the rate

$${\rho }_n(f;D)<\frac{C}{n^p},$$

then $f\left(z\right)\in \mathcal{Z}(\overline{D};\frac{p}{2})$.

Indeed, the inequality $d(\xi ;\alpha )>{\alpha }^2$ follows for convex regions and polygons from Lindelöf’s principle, since there always exists a segment located in $\overline{D}$ and having one of its ends at the boundary point $\xi$; it is enough now to compare the lines of the external level of D and the complement to the mentioned segment.

Note that these theorems apply not only to Jordan domains, but to any domains of the Carathéodory class.

3. Estimation of the Rate of Approximation for Domains with a Smooth Boundary

Theorem 3.1. 

If the domain D is bounded by a smooth curve and $f\left(z\right)$ is analytic in D and continuous in $\overline{D}$, and the $k^{th}$ derivative of the function $f\left(z\right)$ satisfies a Lipschitz condition of order α, $0<\alpha \le 1$, in $\overline{D}$, then for any $\epsilon >0$,

$${\rho }_n(f;D)<\frac{const}{n^{k+\alpha -\epsilon }},$$

(3.1)

where const does not depend on n.

Now let G be a bounded domain with a simply connected complement, whose projection onto the $OJ$ axis is greater than one, and $f\left(z\right)$ be regular in G and continuous in $\overline{G}$. $G_{1+\lambda }$ is the domain bounded by the outer line of level ${\mathcal{Z}}_{1+\lambda }$ of the domain G; ${\pi }_4\left(z\right)$ and ${\pi }_5\left(z\right)$ are two polynomials of degree n with the following properties:

(1)

$|{\pi }_4\left(z\right)|<M$ in the part ${\overline{G}}_{1+\lambda }$, which is located above the straight line $y=c$; similarly $|{\pi }_5\left(z\right)|<M$ in the part $G_{1+\lambda }$, located below the straight line $y=d$; we assume that the segment $(c;d)$ belongs to the projection of G onto the axis $OJ$, and $d-c=1$.

(2)

In the part G located in the half-plane $y\le d$,

$$|{\pi }_5\left(z\right)-f\left(z\right)|<\frac{C_7}{n^{\alpha -\epsilon }};$$

in the part G located in the half-plane $y\ge c$,

$$|{\pi }_4\left(z\right)-f\left(z\right)|<\frac{C_7}{n^{\alpha -\epsilon }}.$$

Let us additionally assume that the distance of any point ${\mathcal{Z}}_{1+\lambda }$ to $\overline{G}-G$ is less than $C_8{\lambda }_{1-\epsilon }$.

Lemma 3.1. 

There exists a polynomial ${\pi }_6$ of degree n for which

1. 

in the domain $G_{1+\frac{\lambda }{2}}$

$$|{\pi }_6\left(z\right)|<2M,$$

2. 

and in the domain G

$$|{\pi }_6\left(z\right)-f\left(z\right)|<\frac{C_9}{n^{\alpha -3\epsilon }}.$$

If $\omega \left(\delta \right)$ denotes the modulus of continuity of $f\left(z\right)$ in D, then the following proposition can be proven in a completely analogous manner.

Theorem 3.2. 

For any $\epsilon >0$ there is a constant C such that

$${\rho }_n(f;D)<C\omega \left(\frac{1}{n^{1-\epsilon }}\right).$$

(3.2)

Let us now consider the connection between the rate of best approximation and the properties of functions under some additional restrictions on the smoothness of the boundary.

Let $\gamma \left(\delta \right)$ denote the modulus of continuity of the function $z^{\prime }\left(s\right)$ ($z\left(s\right)$ is the parametric equation of the boundary; s is the length of the arc $\Gamma$).

Theorem 3.3. 

If $f\left(z\right)\in \mathcal{Z}(\overline{D};p)$ and

$${\int }_0^a\frac{\gamma \left(x\right)}{x}dx<\infty ,$$

(3.3)

then

$${\rho }_n(f;D)<const{\left(\frac{\ln n}{n}\right)}^p;$$

(3.4)

if

$${\int }_{\epsilon }^a\frac{\gamma \left(x\right)}{x}dx>\left|\ln \ln \epsilon \right|\left|\ln \ln \ln \epsilon \right|,$$

(3.5)

then, generally speaking, there is a function $f\left(z\right)\in \mathcal{Z}(D;p)$ such that

$$\underset{n\to \infty }{lim}sup\frac{{\rho }_n(f;D)n^p}{{\left(\ln n\right)}^p}=\infty .$$

(3.6)

Theorem 3.4. 

If $\lambda =1$, then, generally speaking, for some $f\left(z\right)\in \mathcal{Z}(D;\alpha )$

$${\rho }_n(f;D)>const{\left(\frac{{\ln }_{s}n}{n}\right)}^{\alpha };$$

if $\lambda <1$, then for some $f\left(z\right)\in \mathcal{Z}(D;\alpha )$ the following inequality holds:

$${\rho }_n(f;D)>const{\left\{\frac{exp\frac{1}{1-\lambda }{\left({\ln }_{s}n\right)}^{1-\lambda }}{n}\right\}}^{\alpha }.$$

Let (3.3) hold and

$${\rho }_n(f;D)<\frac{const}{n^{k+\alpha }},$$

(3.7)

where k is an integer, $0<\alpha \le 1$.

Theorem 3.5. 

If $\alpha <1$, then $f\left(z\right)\in \mathcal{Z}(D;k+\alpha )$; if $\alpha =1$, then this conclusion, as is known, is not true for analytical domains. In the case of $\alpha =1$, in order for $f^{\left(k\right)}\left(z\right)$ to satisfy a first-order Lipschitz condition, it is necessary and sufficient that

$$\sum _{n=1}^{\infty }{\rho }_n(f;D)n^k<\infty .$$

(3.8)

Let us note the following proposition, the proof of which we will not dwell on.

Lemma 3.2. 

Let ${\alpha }_1\ge {\alpha }_2\ge {\alpha }_3\cdots \ge {\alpha }_n\ge \cdots \to 0$ be an arbitrary sequence of numbers monotonically decreasing towards zero, and $n_1<n_2<\cdots <n_k<\cdots$ be an increasing sequence of integers.

In order for, from the convergence of the series

$$\sum _{i=1}^{\infty }{\alpha }_i,$$

the convergence of the series

$$\sum _{k=1}^{\infty }{n}_k{\alpha }_{n_k}$$

would always follow, it is necessary and sufficient that there exists a positive number æ such that for all k, starting from a sufficiently large one,

$$\frac{n_{k+1}}{n_k}>1+\ae .$$

In particular, it follows that

$$\sum _{n=1}^{\infty }{2}^{n(k+1)}{\rho }_{2^n}(f;D)<\infty ,$$

that is, $f\left(z\right)$ satisfies a first-order Lipschitz condition.

As for the general case of domains with a smooth boundary that do not satisfy condition (3.3), then $f\left(z\right)\in \mathcal{Z}(D;p)$ giving an estimate ${\rho }_n(f;D)$ that relates to the entire class of domains with smooth boundaries and is better than

$${\rho }_n(f;D)<\frac{C\left(\epsilon \right)}{n^{p-\epsilon }},$$

is impossible, since the following can be proven:

Proposition 3.1. 

Let $\psi \left(\delta \right)$ be a monotone function decreasing to zero for $\delta \to +0$ and for any $\epsilon >0$ satisfying the condition

$$\underset{\delta \to 0}{lim}\frac{{\delta }^{\epsilon }}{\psi \left(\delta \right)}=0.$$

There is a domain $D_1$ with a smooth boundary such that the rate of approximation $n^{-k-\alpha }\left(0<\alpha \le 1\right)$ guarantees for some boundary points ζ the inequality

$$|f^{\left(k\right)}\left(z^{\prime }\right)-f^{\left(k\right)}\left(z^{″}\right)|<\psi (|z^{\prime }-z^{″}\left|\right)|z^{\prime }-z^{″}{|}^{\alpha }·const$$

($z^{\prime }$ and $z^{″}$ belong to ${\overline{B}}_{\zeta }$) and there is another domain $D_2$, also with a smooth boundary, and a function $f\left(z\right)$ such that ${\rho }_n(f;D_2)<\frac{1}{n^{k+\alpha }}$; however, for some boundary points ζ

$$\underset{z^{\prime },z^{″}\in {\overline{B}}_{\zeta }}{sup}|f^{\left(k\right)}\left(z^{\prime }\right)-f^{\left(k\right)}\left(z^{″}\right)|>const\frac{|z^{\prime }-z^{″}{|}^{\alpha }}{\psi \left(\right|z^{\prime }-z^{″}\left|\right)}.$$

4. This Section Is Missing from the Manuscript

As we noted above, Section 4 is indicated in the Introduction, but Section 4 is missing from the manuscript. The material intended for Section 4 is set out in Section 3.

5. Some Quasi-Analytic Classes of Functions

Academician S.N. Bernstein in 1923 showed [7] that the class of functions defined on $[0;1]$, for which

$$E_n\left(f\right)<q^n,n=n_1,n_2,n_3,\dots ,n_k,\dots ,$$

where $n_1<n_2<n_3<\cdots <n_k<\dots$ is a particular sequence of integers, is a quasi-analytic class, in the sense that if any two of its functions coincide on any part of the interval $[0;1]$, then they are identical.

Below we give some other quasi-analytic classes of functions defined using best approximations.

Let $\phi \left(n\right)$ be a positive function of the integer argument n. The class of functions for which

$$E_n\left(f\right)<\phi \left(n\right),n=n_1,n_2,\dots ,$$

where $n_1<n_2<n_3<\cdots <n_k<\dots$ is some sequence of integers, is denoted by $Q_{\phi \left(n\right)}$.

Let $\psi \left(\delta \right)$ be a monotone function satisfying the condition

$$\underset{\delta \to 0}{lim}\frac{\psi \left(\delta \right)}{{\delta }^n}=0,n=1,2,3,\dots ,$$

where ${\delta }_n$ means the root of the equation $\psi \left(\delta \right)={\delta }^n{C}^n$ for some fixed $C>0$.

Theorem 5.1. 

If

$$\phi \left(n\right)=O\left({{\delta }_n}^n\right),$$

then the class $Q_{\phi \left(n\right)}$ is quasi-analytic in the sense that if with respect to any two of its functions $f_1\left(x\right)$ and $f_2\left(x\right)$ it is known that

$$|f_1\left(x\right)-f_2\left(x\right)|<\psi (|x-x_0\left|\right),$$

where $x_0\in [0;1]$, then $f_1\left(x\right)\equiv f_2\left(x\right)$.

If, in particular, a function $f\left(x\right)$ of class $Q_{\phi \left(n\right)}$ decreases around some point as $\psi \left(\delta \right)$, then $f\left(x\right)\equiv 0$ is required.

Let $\mathcal{F}$ be a closed bounded set that does not break up the plane, and M be an infinite set of points belonging to $\mathcal{F}$ ($\mathcal{F}$ is assumed to be infinite).

We denote by ${\mathcal{U}}_{\phi \left(n\right)}$ the class of functions continuous on $\mathcal{F}$ for which

$${\rho }_n(\mathcal{F};f)<\phi \left(n\right),n=n_1,n_2,\dots .$$

Theorem 5.2. 

For any infinite set M there is a positive function ${\psi }_M\left(n\right)$ for which the class ${\mathcal{U}}_{{\psi }_M\left(n\right)}$ is quasi-analytic in the sense that from the coincidence of any two of its functions $f_1\left(z\right)$ and $f_2\left(z\right)$ on the set M their identity on $\mathcal{F}$ follows.

Let $\omega \left(\delta \right)$ be the modulus of continuity of $f\left(z\right)$ in $\left|z\right|\le 1$. Suppose that the zeros of $f\left(z\right)$ located on $\left|z\right|=1$ are condensed to the point $z=1$. To characterize the rate of condensation to the limit point, we introduce the function $\lambda \left(r\right)$; ${\lambda }_1\left(\phi \right)$ is the distance $e^{i\phi }$ to M; if $e^{ir}\notin M$, then

$$\lambda \left(r\right)=\underset{\phi \le r}{max}{\lambda }_1\left(\phi \right).$$

If $e^{ir}\in M$, then

$$\lambda \left(r\right)=\underset{\phi \to r}{lim}\lambda \left(\phi \right).$$

Theorem 5.3. 

If

$$\lambda \left(r\right)<{\omega }^{-1}\left(e^{-\gamma \left(r\right)}\right)\mathrm{and}{\int }_0^a\gamma \left(r\right)dr=+\infty ,$$

then $f\left(z\right)\equiv 0$.

Theorem 5.4. 

If $f\left(z\right)$ can be approached in $f\left(z\right)$ with the rate

$${\rho }_n\left(f\right)<e^{-\frac{n}{\ln n}},$$

(5.1)

then the cardinality of the set of zeros $f\left(z\right)$ is at most ${\aleph }_{\mathcal{0}}$.

Remark 5.1. 

From the proof it is clear that more can be stated, namely, if $f\left(z\right)$ is regular in $\left|z\right|<1$, and

$${\rho }_n(f;|z|\le 1)<e^{-\frac{n}{\ln n}},$$

then $f\left(z\right)$ has only a finite number of zeros in $\left|z\right|\le 1$.

6. Best Approximation on Closed Sets

Let E be a closed set of points located in the plane of the complex variable z, and let $f\left(z\right)$ be a function defined and continuous on E. We denote by $\rho \left(n\right)$ the infimum of the numbers

$$\underset{z\in E}{max}|f\left(z\right)-{\mathcal{P}}_n\left(z\right)|$$

by any polynomials of degree $\le n$.

Suppose that E is bounded, is not dense anywhere, and does not break up the plane; according to Lavrentiev’s theorem [8], $\rho \left(n\right)\to 0$ always for $n\to \infty$.

Theorem 6.1. 

Let $M\left(r\right)$ be an arbitrary function growing to $+\infty$ as $r\to \infty$ faster than any power of r, i.e.,

$$\underset{r\to \infty }{lim}\frac{M\left(r\right)}{r^n}=\infty ,n-\mathrm{any},$$

and E be an unbounded set.

There is a function $\phi \left(r\right)>0$ such that from the inequality

$$\rho \left(n\right)<\phi \left(n\right),n=1,2,3,\dots ,$$

it follows that the function $f\left(z\right)$ can be continued from the set E to the entire plane so that the function $\mathcal{F}\left(z\right)$ obtained as a result of the continuation will be an entire function satisfying the condition

$$\underset{\left|z\right|=r}{max}\left|\mathcal{F}\left(z\right)\right|<M\left(r\right).$$

Theorem 6.2. 

Whatever the functions $\phi \left(n\right)$ and $\omega \left(\delta \right)$, $\omega \left(\delta \right)\to 0$ as $\delta \to 0$, there exists a set $\mathcal{P}\subseteq [0,1]$ and $f\left(x\right)$ so that

$${\rho }_n(f;\mathcal{P})<\phi \left(n\right),n=1,2,3,\dots ;$$

however,

$${\omega }_{a,b}\left(\delta \right)>w\left(\sigma \right),\delta <\delta (a;b),$$

where ${\omega }_{a,b}\left(\delta \right)$ is the modulus of continuity of $f\left(x\right)$ on the portion $\mathcal{P}$ contained in the segment $[a,b]$ (${\omega }_{a,b}\left(\delta \right)=1$, if $\mathcal{P}[a,b]=0$).

The above two qualitative results determine the formulation of the problem of studying the best approximation on closed sets depending on the properties of these sets and on the behavior of the approximated functions on them.

Without dwelling here on all sorts of problems in this direction, let us consider one of them.

According to a well-known classical result, if $\mathcal{P}$ is a segment or closed domain and the function $f\left(z\right)$ can be approached with a progression rate on $\mathcal{P}$, i.e.,

$${\rho }_n(f;\mathcal{P})<q^n,0<q<1,$$

then $f\left(z\right)$ is analytic at every point of $\mathcal{P}$.

Let us consider the question for which, more generally, set $\mathcal{P}$ the previous result continues to be valid.

Theorem 6.3. 

Suppose the set $\mathcal{P}$ is perfect, with the connected complement, and for any function $\phi \left(z\right)$ continuous on $\mathcal{P}$, there exists a function $u\left(z\right)$ harmonic on the complement of $\mathcal{P}$, taking at the point $\mathcal{P}$ value $\phi \left(z\right)$. If for some function $f\left(z\right)$ defined on $\mathcal{P}$

$$\rho \left(n\right)<q^n,0<q<1,n=1,2,3,\dots ,$$

(6.1)

then $f\left(z\right)$ is analytic at every point of $\mathcal{P}$.

Remark 6.1. 

Theorem 6.3 can also be formulated in local form: if a given point $z_0$ is regular (in the sense of the Dirichlet problem), then (6.1) implies the analyticity of $f\left(z\right)$ at this point.

Remark 6.2. 

Thus, the possibility of extending the above-mentioned classical result to a more general type of set is influenced not by the measure (linear or flat), but by a more subtle characteristic of the set—its capacity.

In the case where $\mathcal{P}$ has only one limit point, the capacity of $\mathcal{P}$ is zero; hence, the previous theorem says nothing.

Theorem 6.4. 

For any positive function $\phi \left(n\right)$ satisfying the condition

$$\underset{n\to \infty }{lim}\frac{\ln \phi \left(n\right)}{n}=-\infty ,$$

(6.2)

there exists a countable set E on $[0;1]$, having only one limit point and such that if the inequality

$$\rho \left(n\right)<\phi \left(n\right)$$

holds for some function $f\left(x\right)$ defined on E, it follows that $f\left(x\right)$ takes its values from some entire function $\mathcal{F}\left(z\right)$.