MATHEMATICS

L. I. RONKIN

ON QUASIANALYTIC CLASSES OF FUNCTIONS OF SEVERAL VARIABLES

(Presented by Academician S. N. Bernstein on 17 IV 1962)

This article adjoins the joint work of V. I. Matsaev and the author ((^1)), in which some generalizations of the well-known Carleman theorem on quasianalytic classes were obtained for the case of functions of any finite number of independent variables.

1. Following ((^1)), we shall call a certain class of functions quasianalytic I if it contains no function, different from identically zero, that vanishes at some point together with all its derivatives. We shall call a class of functions quasianalytic II if it contains no nontrivial finite function, i.e., a function different from identically zero and equal to zero everywhere outside some finite domain.

Let (D) be a certain domain in the plane (R_2) of the real variables (x, y)*. Further, let ({m_{p,q}}) be a certain double sequence of nonnegative numbers. Denote by (C_D(m_{p,q})) the totality of all functions infinitely differentiable in the domain (D) for which

[
\sup_{(x,y)\in D}\left|\frac{\partial^{p+q} f(x,y)}{\partial x^p \partial y^q}\right| < M r^p s^q m_{p,q},
]

where the numbers (M, r, s) are specific to each function of the class (C_D(m_{p,q})). This definition is analogous to the definition of the class (C(m_n)) of functions of one variable. Let us note that, as was shown by S. Mandelbrojt ((^2)), for the classes (C(m_n)), quasianalyticity I and II are equivalent. In the case of functions of several variables there is no such equivalence. This was shown in ((^1)) by comparing the conditions for quasianalyticity II of the class (C_D(m_{p,q})), obtained earlier by P. Lelong ((^3)), and the conditions for quasianalyticity I of the class (C_D(m_{p,q})), obtained in ((^1)).

1. In the case when (D = R_2), the following was obtained in ((^1)).

Theorem 1. For quasianalyticity I of the class (C_D(m_{p,q})) it is necessary and sufficient that each of the sequences ({m_{p,0}}), ({m_{0,q}}) generate a quasianalytic class of functions of one variable.

We shall prove here this theorem for the case of an arbitrary domain (D). For this we shall need the following lemmas:

Lemma 1. Let the function (f(x)) be infinitely differentiable on the interval ([-1,1]). Put (\psi(x)=f(\cos x)). Then, if

[
\sup_{-1\le x\le 1}\left|f^{(p)}(x)\right| < M_p,\qquad p=0,1,2,\ldots,
]

then

[
\sup_{-\infty<x<\infty}\left|\psi^{(p-1)}(x)\right| < c^p M_p + 4M_0 p^p,
]

where (c) is a constant independent of the function.

* Consideration of a larger number of variables differs from that set out in §§ 1 and 2 only by a complication of notation.

For the proof of the lemma we use a device which was applied by S. N. Bernstein ((^{4})) in proving his theorem on the influence of the rate of approximation of a function by polynomials on its differential properties.

Denote by (B_n(x)) the trigonometric polynomial of degree (n) which deviates least from the function (\psi(x)), and by (s_n(x)) the partial sum of the Fourier series of the function (\psi(x)). Also put
[
u_k(x)=B_{2^{k+1}p}(x)-B_{2^k p}(x).
]
Then
[
\psi(x)=s_p(x)+\bigl(B_{2p}(x)-s_p(x)\bigr)+\sum_{k=1}^{\infty}u_k(x).
]

Let us estimate the terms of the series in modulus. For (k\geqslant 1) and any (x\in(-\infty,\infty)) we have
[
|u_k(x)|\leqslant |B_{2^{k+1}p}(x)-\psi(x)|+|B_{2^k p}(x)-\psi(x)|
\leqslant 2E^T_{2^k p},
]
where
[
E^T_n=\max_{-\infty<x<\infty}|\psi(x)-B_n(x)|.
]

It is known (see, for example, ((^{5})), p. 157) that (E^T_n=E_n), where (E_n) is the best approximation of the function (f(x)) by algebraic polynomials of degree not exceeding (n). On the other hand, by Jackson’s theorem (see, for example, ((^{5}))), beginning with (n\geqslant p), we have
[
E_n<c_1^p M_p n^{-p},
]
where the constant (c_1<24e). Consequently,
[
\max_{-\infty<x<\infty}|u_k(x)|\leqslant 2c_1^p p^{-p}2^{-kp}M_p.
]

From the same considerations, and also by virtue of Lebesgue’s theorem (see, for example, ((^{5,6}))) on the approximation of a periodic function by partial sums of its Fourier series, we obtain
[
\max_{-\infty<x<\infty}|B_{2p}(x)-s_p(x)|
\leqslant 2\max_{-\infty<x<\infty}|\psi(x)-s_p(x)|
\leqslant
]
[
\leqslant 2(3+\ln p)E^T_p\leqslant 2(3+\ln p)c_1^p p^{-p}M_p.
]

Since each coefficient of the Fourier series of the function (\psi(x)), obviously, does not exceed (2M_0), it follows that (|s_p(x)|\leqslant 4pM_0) for any (x).

We now estimate (\psi^{(p-1)}(x)). By S. N. Bernstein’s inequality for trigonometric polynomials, for any (x\in(-\infty,\infty)) we have
[
|\psi^{(p-1)}(x)|\leqslant |s_p^{(p-1)}(x)|
+\left|\bigl(B_{2p}(x)-s_p(x)\bigr)^{(p-1)}\right|
+\sum_{k=1}^{\infty}|u_k^{(p-1)}(x)|\leqslant
]
[
\leqslant 4p^pM_0+2^{p+1}c_1^pM_p
+\sum_{k=1}^{\infty}(p2^{k+1})^{p-1}p^{-p}2^{-pk}M_p
\leqslant 4p^pM_0+c^pM_p.
]
The lemma is proved.

Lemma 2. If the class (C(m_n)) is quasianalytic, then the class (C(m_n+n^n)) is also quasianalytic *.

Proof. Let (C(m_n)) be a quasianalytic class. Two cases are possible: either (m_n\geqslant n^n) for all (n), starting from some (n), and then the assertion of the lemma is obvious; or there exists such an infinite

* It is known ((^{7})) that the sum of two functions from different quasianalytic classes may turn out to be a function belonging to no quasianalytic class. At the same time, by Lemma 2, it follows that the sum of a function from a quasianalytic class and an analytic function is a function from a quasianalytic class.

a sequence of indices (n_k), such that (m_{n_k}2n_k) for every (k). Put

[
\beta_n=\inf_{k\ge n}\sqrt[k]{m_k+k^k}.
]

By Carleman’s theorem (see, for example, (8), p. 104) on quasianalytic classes of functions, in order to prove the lemma it is enough to show that

[
\sum_{n=1}^{\infty}\frac1{\beta_n}=\infty.
]

Observe that (\beta_{n_k}\le n_k\sqrt[n_k]{2}<2n_k), and that (\beta_{n_1}\ge \beta_{n_2}) for (n_1\le n_2). Consequently,

[
\sum_{n=1}^{\infty}\frac1{\beta_n}
=
\sum_{k=0}^{\infty}\sum_{i=n_k+1}^{n_{k+1}}\frac1{\beta_i}
\ge
\sum_{k=0}^{\infty}\frac{n_{k+1}-n_k}{n_{k+1}}
=\infty.
]

The lemma is proved.

Now let (f(x,y)\in C_D(m_{p,q})) vanish, together with all its derivatives, at the point ((x_0,y_0)), and let the sequences ({m_{p,0}}) and ({m_{0,q}}) generate quasianalytic classes of functions of one variable. Without loss of generality one may assume that (D) is the square ({-1\le x,y\le 1}) and that (x_0=0,\ y_0=0). Consider the function (\psi(x,y)=f(\cos x,\cos y)). It is easy to see that this function belongs to some class (C_{R_2}(M_{p,q})), for which, by Lemma 1, (M_{p-1,0}\le 4p^p m_{0,0}+c^p m_{p,0}) and (M_{0,q-1}\le 4q^q m_{0,0}+c^q m_{0,q}). By Lemma 2 the sequences ({M_{p,0}}), ({M_{0,q}}) generate quasianalytic classes of functions of one variable.* Since the function (\psi(x,y)), together with all its derivatives, vanishes at the point ((\pi/2,\pi/2)), it follows from Theorem 1, proved in (1) for the case (D=R_2), that (\psi(x,y)\equiv 0). Hence (f(x,y)\equiv 0).

Necessity is verified in exactly the same way as in the case (D=R_2).

1. P. Lelong (3) showed that for quasianalyticity of the class II (C_{R_2}(m_{p,q})) it is necessary and sufficient that the sequence

[
l_n=\min_{p+q=n}{m_{p,q}}
]

generate a quasianalytic class of functions of one variable. By the definition of a quasianalytic class II, a nontrivial function from it cannot be equal to zero everywhere outside a finite domain. However, it may be equal to zero everywhere inside some domain. It turns out that such domains of zeros cannot be arbitrary; more precisely, the following theorem holds.

Theorem 2. Let (L) be some curve, and let (D) be the smallest rectangle with sides parallel to the coordinate axes that contains (L). Suppose further that (f(x,y)\in C_D(m_{p,q})), where the class (C_D(m_{p,q})) is quasianalytic II. Then (f(x,y)=0) for ((x,y)\in D).

The proof of this theorem is based on the following lemmas:

Lemma 3. If (|f^{(p)}(x)|\le M_p) for (x\in[0,a]), (p=0,1,2,\ldots), and (f^{(p)}(\xi)=0), where (\xi\in[0,a]), (p=0,1,2,\ldots), then for (x\in[-\sqrt a,\sqrt a])

[
\left|[f(x^2)]^{(p)}\right|\le (2\sqrt a\, e^{\sqrt2})^p M_p.
]

Lemma 4. Suppose the function (f(x,y)) satisfies the hypotheses of Theorem 2, with (D={0\le x\le a,\ 0\le y\le b}). Then

[
f(x^2,y)\in C_{D_1}(m_{p,q}),\qquad
f(x,y^2)\in C_{D_2}(m_{p,q}),\qquad
f(x^2,y^2)\in C_{D_3}(m_{p,q}),
]

* Estimates for the derivatives of the functions (\psi(x)=f(\cos x)), different from those obtained in Lemma 1, are available in (8). Relying on these estimates, we were not able to show that from the membership of (f(x)) in a quasianalytic class there follows the membership of (\psi(x)) in some quasianalytic class.

where (D_1, D_2, D_3) are respectively the rectangles
({|x|\leq \sqrt a,\ 0\leq y\leq b}),
({0\leq x\leq a,\ |y|\leq \sqrt b}),
({|x|\leq \sqrt a,\ |y|\leq \sqrt b}).

The proof of Lemma 3 essentially repeats the proof of one lemma of Mandelbrojt ((2), p. 24).

Lemma 4 is easily proved with the aid of Lemma 3.

Let us outline the proof of the theorem. Suppose that there exists a point ((x_1,y_1)) at which (f(x_1,y_1)\ne 0). Then, with the aid of Lemma 4, one constructs a function (\psi(x,y)\in C_{D^}(m_{p,q})), where (D^) is one of the rectangles (D_1,D_2,D_3), which vanishes together with all its derivatives on some closed curve (L^) and is nonzero at least at one point inside the domain (K) bounded by the curve (L^). But then the function equal to (\psi(x,y)) for ((x,y)\in K) and equal to zero for ((x,y)\notin K) also belongs to the class (C_{D^}(m_{p,q})) and is a nontrivial finite function, which contradicts the quasi-analyticity II of the class (C_{D^}(m_{p,q})). The theorem is proved.

Let us note a simple but, in our opinion, interesting consequence. Let the curve (L) intersect every straight line parallel to either of the coordinate axes. Then, in order that the class (C_{R_2}(m_{p,q})) contain no nontrivial function vanishing on the curve (L) together with all its derivatives, it is necessary and sufficient that the class (C_{R_2}(m_{p,q})) be quasi-analytic II.

It can also be proved that if the curve (L) is finite, then, in order that in the class (C_{R_2}(m_{p,q})) there be no nontrivial function equal to zero together with all its derivatives on (L), it is necessary and sufficient that the class (C_{R_2}(m_{p,q})) be quasi-analytic I.

Kharkov
State University

Received
12 IV 1962

References Cited

1. V. I. Matsaev, L. I. Ronkin, Zap. matem. otd. fiz-matem. fak. i Kharkovsk. matem. obshch., 27, ser. 4 (1961).
2. S. Mandelbrojt, Quasi-analytic Classes of Functions, 1937.
3. P. Lelong, C. R., 232, 1178 (1951).
4. S. N. Bernstein, Soobshch. Kharkovsk. matem. obshch., 13, ser. 2 (1912); Collected Works, 1, 1952, p. 11.
5. I. P. Natanson, Constructive Theory of Functions, 1949.
6. N. I. Akhiezer, Lectures on the Theory of Approximation, 1947.
7. S. Mandelbrojt, Acta Math., 72, 15 (1940).
8. S. Mandelbrojt, Adjacent Series, Regularization of Sequences, Applications, IL, 1955.