MATHEMATICS

E. P. DOLZHENKO

ON THE DIFFERENTIATION OF COMPLEX FUNCTIONS

(Presented by Academician A. N. Kolmogorov, 10 IX 1959)

§ 1. Let (E) be some set in the complex plane; (\zeta) a limit point of this set, (\zeta \in E); (f(z)) a complex function (which may also take infinite values) defined on (E).

A number (a) (finite or not) is called a derivative number of the function (f(z)) at the point (\zeta) (with respect to the set (E)) if there exists a sequence of complex numbers ({z_n}) such that: 1) (z_n \in E) ((n=1,2,\ldots)), (z_n \to \zeta) as (n \to \infty); 2) all the differences (f(z_n)-f(\zeta)) have meaning (i.e., for any (n) at least one of the numbers (f(z_n)), (f(\zeta)) is different from infinity); 3)

[
a=\lim_{n\to\infty}\frac{f(z_n)-f(\zeta)}{z_n-\zeta},
]

if (\zeta \ne \infty), or

[
a=\lim_{n\to\infty} z_n [f(z_n)-f(\zeta)],
]

if (\zeta=\infty).

Let (\mathfrak{M}_f(\zeta)) be the set of all derivative numbers of the function (f(z)) at the point (\zeta \in E) with respect to (E).

Theorem 1. For an arbitrary complex function (f(z)) (finite or not), defined on some set () (E) of the complex plane, at almost every point () (\zeta \in E) one of the following four cases holds:*

1) (\mathfrak{M}_f(\zeta)) is some circle (|z-a(\zeta)|=r(\zeta)), (0 \le r(\zeta)<\infty) (in particular, when (r(\zeta)=0), (\mathfrak{M}_f(\zeta)) consists of a single point, i.e. in this case (f(z)) has a derivative at the point (\zeta));

2) (\mathfrak{M}_f(\zeta)) coincides with the (extended) complex plane;

3) (\mathfrak{M}_f(\zeta)) consists of some circle (|z-a(\zeta)|=r(\zeta)), (0 \le r(\zeta)<\infty), and the point (z=\infty);

4) (\mathfrak{M}_f(\zeta)) consists of the single point (z=\infty).

Remark 1. At almost every point of finiteness of (f(z)), one of the first three cases holds.

Remark 2. At almost every point of discontinuity of (f(z)), one of the first two cases holds.

Remark 3. At points (\zeta \in E) at which the real and imaginary parts of (f(z)) have total differentials (with respect to the set (E)), case 1) holds; conversely, almost everywhere where case 1) holds, the real and imaginary parts of (f(z)) have a total differential (with respect to (E)).

Remark 4. Each of the four cases may hold on a set of positive measure (concerning the occurrence of the first two of them, see ((^{2,3}))).

The theorem just formulated is an analogue of the well-known theorem of Denjoy ((^1)) on derivative numbers of a real function of a real variable. It generalizes a result of Yu. Yu. Trokhimchuk ((^2)), obtained for functions continuous in a domain.

* Generally speaking, nonmeasurable.

** That is, with the possible exception of points (\zeta \in E) which together form a set of measure zero.

§ 2. Let, as before, (E) be some set in the complex plane, but (\infty\notin E). Let, further, (f(z)) be a complex function, defined and finite on (E), and let (\zeta\in E) be a limit point of (E). We say that (f(z)) has (k) Vallee-Poussin derivatives at the point (\zeta) (with respect to the set (E)) if, for (\zeta+h\in E),

[
f(z+h)=f(z)+h f_{(1)}(z)+\frac{h^2}{2!} f_{(2)}(z)+\cdots+\frac{h^k}{k} f_{(k)}(z)+o(h^k).
]

Here the number (f_{(i)}(z)) ((1\le i\le k)) (determined, obviously, uniquely) is called the (i)-th Vallee-Poussin derivative (the (i)-th ((V!-!P))-derivative).

The advantage of ((V!-!P))-differentiation over ordinary, successive differentiation consists in the fact that the (k)-th ((V!-!P))-derivative is defined “at a point,” independently of whether or not the preceding ((V!-!P))-derivatives exist in some neighborhood of this point. Note that the first ((V!-!P))-derivative coincides with the ordinary first derivative, and that from the existence on the set (e\subset E) of (k) ((V!-!P))-derivatives it follows that the function (f(z)) is approximately differentiable (k) times almost everywhere on (e).

Below (R_n[f]) denotes the best approximation to the function (f(z)) by rational functions of order not exceeding (n)* (i.e. (R_n[f]=\inf_\varphi{\sup_{z\in E}|f(z)-\varphi(z)|}), where (\varphi) ranges over all rational functions of order not exceeding (n)); (R_n(z)) denotes a rational function of order not exceeding (n) which realizes this approximation ((R_n(z)) exists ({}^{(4)})).

Theorem 2. If, for a function (f(z)) defined on some set (E),

[
R_n[f]\le \frac{C}{n^{p+\varepsilon}},
]

where (C<\infty) does not depend on (n), (p) is natural, and (\varepsilon>0), then almost everywhere— with respect to planar measure if (E) is planar, and with respect to linear measure if (E) belongs to the real axis—(f(z)) has (p) Vallee-Poussin derivatives, and

[
f_{(i)}(z)=\lim_{k\to\infty} R'_{n_k}(z)\qquad (1\le i\le p)
]

almost everywhere (with respect to the corresponding measure) on (E), whatever lacunary sequence ({n_k}) may be.

In the case where (E=[0,1]), this theorem substantially strengthens Theorem 3 of A. A. Gonchar’s paper ({}^{(5)}), and also contains Theorem 2 of that paper.

Theorem 3. Let a sequence of nonnegative numbers ({a_n}) be nonincreasing, and let (E) be a set of points on the real axis. In order that every function (f(x)) ((x\in E)) for which (R_n[f]\le a_n) be differentiable almost everywhere on (E) (with respect to (E)), it is necessary and sufficient that

[
\sum_{n=1}^{\infty} a_n<\infty .
\tag{1}
]

In general, more can be proved, namely:

If (R_n[f]\le a_n) and (1) is satisfied, then (f(x)) coincides on (E) with some function (f^(x)), absolutely continuous on the entire line ((-\infty,\infty)), and almost everywhere on (E)*

[
f'(x)=\lim_{k\to\infty} R'_{n_k}(x)
]

for any lacunary sequence of numbers (n_k).

* The order of a rational function (\varphi(z)=\dfrac{a_m z^m+\cdots+a_0}{b_k z^k+\cdots+b_0}) is called (n=\max{m,k}).

** It is possible that the condition (\sum_{n=1}^{\infty} a_n<\infty) is not only sufficient for the absolute continuity of (f(z)), but also necessary.

If, however,

[
\sum_{n=1}^{\infty} a_n=\infty,
]

then on (E) there exists a function (f_1(x)), nowhere on (E) (with respect to (E)) differentiable, and a function (f_2(x)), almost nowhere on (E) approximately differentiable, such that (R_n[f_i]\leq a_n) ((i=1,2)). This fact is of fundamental importance, since it says that the minimal conditions on (R_n[f]) ensuring approximate differentiability (almost everywhere) coincide with the minimal conditions on (R_n[f]) ensuring ordinary differentiability (also almost everywhere), despite the fact that the requirement of approximate differentiability is much weaker than the requirement of ordinary differentiability.

§ 3. Let (K) be a continuum (i.e., a bounded closed connected set) lying in the complex plane (Z) ((K) may be nowhere dense in (Z)). The complement of (K) decomposes into a finite or countable set of nonintersecting simply connected domains (g_i) with boundaries, respectively, (\gamma_i). We shall denote the diameter of (\gamma_i) by (d(\gamma_i)). Consider the set consisting of all functions that have a continuous derivative on (K), and all uniform (on (K)) limits of these functions, and denote it by (A[K]). The space (A[K]) is a Banach space* and, by a theorem of A. G. Vitushkin(^6), coincides with the set of all functions admitting uniform (on (K)) approximation with arbitrary accuracy by rational functions.

In (^7) it is shown that no conditions on the continuum (K) ensure ordinary differentiability of functions from (A[K]) at any point (\xi\in K). However, the following is true:

Theorem 4. Let the continuum (K) be such that

[
\sum_i [d(\gamma_i)]^{\frac{2}{m+2}}
\quad (m\ \text{natural}).
\tag{2}
]

Then (K) can be represented in the form

[
K=G\cup \Gamma,
]

where (\Gamma\supset \bigcup_i \gamma_i,\ \operatorname{mes}\Gamma=\operatorname{mes}\bigcup_i\gamma_i,) and (G) is the limit of an increasing sequence of closed sets (E_n) ((E_n\subset E_{n+1})) possessing the following properties:

1) if (f\in A[K]), then on each set (E_n) the function (f(z)) is (m) times differentiable with respect to this set;

2) if a sequence ({f_k(z)}) of functions from (A[K]) converges uniformly (on (K)) to a function (f(z)), then (f\in A[K]), and on each set (E_n) the derivatives of (f_k(z)) (with respect to the set (E_n)) up to order (m) inclusive converge uniformly to the corresponding derivatives of (f(z)).

This theorem is an analogue of Weierstrass’ theorem on a sequence of functions analytic in a domain (G) and converging uniformly inside the domain.

Let us note that if, instead of (2), one requires somewhat more, namely

[
\sum_i [d(\gamma_i)]^{\frac{1}{m+2}}<\infty,
\tag{2'}
]

then each set (E_n) will consist of a finite number of continua. If the series (2′) converges for every (m>0), then all functions from (A[K])

[
\text{* With norm } |f|=\max_{z\in K}|f(z)|.
]

infinitely differentiable on each (E_n), and from the uniform convergence (on (K)) of ({f_k(z)}) to (f(z)) it follows that (f \in A[K]), and also that, on each (E_n), all derivatives of (f_k(z)) converge uniformly on (E_n) to the corresponding derivatives of (f(z)).

Moscow State University
named after M. V. Lomonosov

Received
9 VII 1959

References Cited

(^{1}) S. Saks, Theory of the Integral, IL, 1949.
(^{2}) Yu. Yu. Trokhimchuk, Uspekhi Mat. Nauk, 11, No. 5 (1956).
(^{3}) A. D. Myshkis, Uspekhi Mat. Nauk, 12, No. 2 (1957).
(^{4}) S. L. Walsh, Interpolation and Approximation, N. Y., 1935.
(^{5}) A. A. Gonchar, DAN, 100, No. 2 (1955).
(^{6}) A. G. Vitushkin, DAN, 123, No. 6 (1958).
(^{7}) E. P. Dolzhenko, DAN, 125, No. 5 (1959).