MATHEMATICS

B. S. MITYAGIN

ON AN INFINITELY DIFFERENTIABLE FUNCTION WITH PRESCRIBED VALUES OF THE DERIVATIVES AT A POINT

(Presented by Academician A. N. Kolmogorov, 17 XII 1960)

1. E. Borel in 1895 showed that, for a given numerical sequence \(a_0,a_1,\ldots\), one can construct* in a neighborhood of zero an infinitely differentiable function \(\psi(t)\) such that \(\psi^{(n)}(0)=a_n,\ n\geqslant 0\). The question arose whether such a function can be constructed in a prescribed class defined by inequalities on the derivatives. For the class of analytic functions, where \(|\psi^{(n)}(t)|\leqslant C^n n^n\), this is obviously possible (by means of the Taylor series). For quasianalytic classes \(C(m_n)\) there is the theorem of Carleman \((^3)\), showing how, from the values of the derivatives \(\psi^{(n)}(0)=\xi_n\), to reconstruct uniquely a function \(\psi(t)\in C(m_n)\); however, not for every numerical sequence \(\xi=(\xi_n,n\geqslant 0)\) such that \(|\xi_n|\leqslant AB^n m_n\) does there exist in the class \(C(m_n)\) a function \(\psi(t)\) such that \(|\psi^{(n)}(t)|\leqslant A_1B_1^n m_n\). The question of the possibility of construction in non-quasianalytic classes remained open. This question was posed to the author by G. E. Shilov.

In the present note, for a given numerical sequence \(a_n,\ n=0,1,\ldots\), admitting the estimate \(|a_n|\leqslant CB^n n^{n\beta}\), \(\beta>1\), for any \(\varepsilon>0\) there is constructed on the interval \([-1,1]\) an infinitely differentiable function \(\psi(t)\) such that
\[ \max_{|t|\leqslant 1}|\psi^{(n)}(t)|\leqslant C_1[(1+\varepsilon)\Gamma_\beta B]^n n^{n\beta},\quad n\geqslant 0,\quad \Gamma_\beta= \]
\[ =(\cos \pi/2\beta)^{-\beta}. \]
Various consequences of this fact are given, as well as its refinements and generalizations.

1. Consider (cf. \((^4)\), Ch. 4) the Banach spaces \(C^{\beta B}\) and \(L_2^{\beta B}\) of all infinitely differentiable functions on the interval \([-1,1]\) such that
   \[ C^{\beta B}=\{\varphi:\ |\varphi|_B=\sup_{n,t}|\varphi^{(n)}(t)|/B^n n^{n\beta}<\infty\} \]
   and
   \[ L_2^{\beta B}=\left\{\varphi:\ |\varphi|_{B,2}=\left(\sum_0^\infty \int_{-1}^{1}|\varphi^{(k)}(t)|^2\,dt/B^{2k}k^{2k\beta}\right)^{1/2}<\infty\right\}; \]
   the space \(L_2^{\beta B}\) is Hilbert. Put \(C^\beta=\bigcup_\beta C^{\beta B}\).

A functional \(f\) on the space \(C^{\beta B}\) (or \(L_2^{\beta B}\)) will be called concentrated at the point \(t_0\), if it vanishes on all functions \(\varphi\in C^{\beta B}\) that are identically equal to zero in some neighborhood of the point \(t_0\). For simplicity, in what follows we shall speak of the point \(t_0=0\). With each functional \(f\) concentrated at zero we associate** the function
\[ \Phi(s)= \]
\[ =f\left\{e^{its}h\left(\frac{t}{\varepsilon}\right)\right\}, \]
where \(h(t)\) is any function*** from \(C^\gamma,\ 1<\gamma<\beta\), equal to zero for \(|t|\geqslant 1\) and to one for \(|t|\leqslant 1/2\). The values \(\Phi(s)\) do not depend on \(\varepsilon\), since the functional \(f\) is concentrated at zero. Note that to the functionals \(\delta_n:\delta_n(\varphi)=\varphi^{(n)}(0)\) there correspond the functions \(\Delta_n(s)=(is)^n\).

Lemma 1****. Let \(f\) be a functional concentrated at zero on the space \(C^{\beta B}\), and let \(\Phi(s)\) be the function corresponding to it. The function \(\Phi(s)=\)

* Other solutions of this question, see \((^2)\).

** This function \(\Phi(s)\) is in fact the Fourier transform of the functional \(f\), cf. \((^6)\), p. 225.

*** On the construction of such functions \(h(t)\), see, for example, \((^5)\), p. 105.

**** An analogous fact was noted in \((^7)\), p. 103, which was also the starting point of our work; our proof is technically simpler.

\[ =\sum_0^\infty f_n s^n \]
is entire, and its Taylor coefficients satisfy the inequalities
\[ |f_n|\leq \frac{C_\delta(1+\delta)^n\|f\|}{B^n(n\cos\pi/2\beta)^{\beta n}} \]
for all \(\delta>0\), where \(C_\delta\) are constants independent of \(n\) and \(f\).

Proof. The family \(\varphi_s(t)=e^{its}h(t)\) of elements of the space \(C^{\beta B}\) depends analytically on the parameter \(s\) for all its complex values; therefore the function
\[ \Phi(s)=f(\varphi_s) \]
is entire.

By the definition of \(\Phi(s)\), for all \(\varepsilon>0\) the estimate
\[ |\Phi(s)|\leq \|f\|\,\|e^{its}h(t/\varepsilon)\|_B \]
holds. By the choice of the function \(h\) we have
\[ |h^{(n)}(t)|\leq CA^n n^{n\gamma},\qquad |h^{(n)}(t/\varepsilon)|\leq C(A/\varepsilon)^n n^{n\gamma}; \]
moreover,
\[ \max_{|t|\leq \varepsilon}|(e^{its})^{(n)}|=e^{\varepsilon|\tau|}r^n, \]
where \(s=\sigma+i\tau,\ r=|s|\). Therefore
\[ \left|\left\{e^{its}h\left(\frac{t}{\varepsilon}\right)\right\}^{(n)}\right| \leq e^{\varepsilon|\tau|}C\sum_{k=0}^n \binom{k}{n}r^k\left(\frac{A}{\varepsilon}\right)^{n-k}(n-k)^{(n-k)\gamma} \leq Ce^{\varepsilon|\tau|}\left(r+\frac{A}{\varepsilon}n^\gamma\right)^n . \]

Choose \(\lambda\) so that \(\gamma<\lambda<\beta\). Then for \(r\leq n^\lambda\) we have
\[ \sup_{r\leq n^\lambda} \frac{\left(r+\frac{A}{\varepsilon}n^\gamma\right)^n}{B^n n^{n\beta}} \leq \sup_n \frac{\left(1+\frac{A}{\varepsilon}\right)^n n^{\lambda n}}{B^n n^{n\beta}} =D<\infty, \]
\[ \sup_{r>n^\lambda} \frac{\left(r+\frac{A}{\varepsilon}n^\gamma\right)^n}{B^n n^{n\beta}} \leq \sup_n \frac{\left(r+\frac{A}{\varepsilon}r^{\gamma/\lambda}\right)^n}{B^n n^{n\beta}} . \]

For any \(\delta>0\), when
\[ r\geq R(\varepsilon,\delta;\gamma/\lambda) \]
the inequality
\[ \left(r+\frac{A}{\varepsilon}r^{\gamma/\lambda}\right)\leq (1+\delta)r \]
is satisfied; hence for such \(r\) we have
\[ \sup_n \frac{\left(r+\frac{A}{\varepsilon}n^\gamma\right)^n}{B^n n^{n\beta}} \leq \sup_n \frac{(1+\delta)^n r^n}{B^n n^{n\beta}} \leq \exp\left\{\frac{\beta}{e}\left(\frac{1+\delta}{B}\right)^{1/\beta}r^{1/\beta}\right\}. \]
(For the last inequality see \((^4)\), p. 204.)

The inequalities obtained above show that, if the constants \(C_{\varepsilon\delta}\) are chosen appropriately, then for all \(\delta>0\) and \(\varepsilon>0\) the inequalities
\[ \|e^{its}h(t/\varepsilon)\|_B \leq C_{\varepsilon\delta}\exp\{\varepsilon|\tau|+Kr^{1/\beta}\} \]
and
\[ |\Phi(s)|\leq \|f\|\,C_{\varepsilon\delta}\exp\{\varepsilon|\tau|+Kr^{1/\beta}\} \]
hold, where
\[ K=\frac{\beta}{e}\left(\frac{1+\delta}{B}\right)^{1/\beta}. \]

Thus the function \(\Phi(s)\) is entire, of first order of minimal type, and on the real axis its order of growth is \(1/\beta<1\) and its type
\[ K=\frac{\beta}{e}\left(\frac{1+\delta}{B}\right)^{1/\beta} \]
in the estimate \(|\Phi(s)|\) does not depend on \(\varepsilon\). Then, by the refined Phragmén–Lindelöf theorem,* in the whole complex plane the estimate
\[ |\Phi(s)|\leq D_1\|f\|\exp\{\Gamma_\beta K r^{1/\beta}\} \]
is valid, where
\[ \Gamma_\beta=(\cos\pi/2\beta)^{-\beta}. \]
The Cauchy inequalities for the Taylor coefficients and the relation (see \((^4)\), p. 259, formula 5)
\[ \inf_r\{r^{-n}\exp[Mr^{1/\beta}]\} \leq C\left(M\frac{e}{\beta}\right)^{\beta n}\left(\frac{1}{n}\right)^{\beta n} \]
give
\[ |f_n|\leq D_\delta\left(\frac{1+2\delta}{B}\Gamma_\beta\right)^n n^{-n\beta}\|f\|,\qquad n\leq 0. \]
The lemma is thereby proved.

1. It is not difficult to verify that, for \(B<B_1<B_2\), the inclusions
   \[ C^{\beta B}\subset L_2^{\beta B_1}\subset C^{\beta B_2} \]
   hold, and the corresponding operators are continuous. Hence it is clear that the assertion of Lemma 1 is also true for the spaces \(L_2^{\beta B}\).

Denote by \(K(m_n)\) the Banach space of sequences
\[ \xi=(\xi_n,\ n\geq 0) \]
with norm
\[ \|\xi\|=\sup_n \frac{|\xi_n|}{m_n}. \]

* See \((^8)\), p. 70, Theorem 22; compare with the corollary on p. 71.

Theorem 1. For every \(\varepsilon>0\) there exists a linear continuous operator
\(L: K(B^n n^\beta)\to C^{\beta,\Gamma_\beta(1+\varepsilon)}\) such that
\((L\xi)^{(n)}(0)=\xi_n,\ n\geq 0\).

Proof. Consider the space \(L_2^{\beta,\Gamma_\beta(B+\varepsilon/2)}=X\) and the closed subspace
\(E_0\subset X'\) of functionals concentrated at zero. By Lemma 1, for \(f\in E_0\) we have
\[ |f_n|\leq D\|f\|\left(\frac{1}{B+\varepsilon/4}\right)^n n^{-n\beta},\quad n\geq 0; \]
therefore, for every \(a=(a_n)\in K(B^n n^\beta)\), the formula
\[ A(f)=\sum_0^\infty (-i)^n f_n a_n \]
defines a linear continuous functional on \(E_0\); indeed,
\[ |A(f)|\leq \sum_0^\infty D\|f\|(B+\frac{\varepsilon}{4})^{-n} n^{-n\beta}B^n n^\beta \|a\| = C_\varepsilon\|f\|\cdot\|a\|. \]
The space \(X'\) is Hilbert; let \(P_0\) be the operator of orthogonal projection in it onto \(E_0\). Put, for \(g\in X'\), \(\hat A(g)=A(P_0g)\), and define the operator \(L:K\to X\) by the formula \(\hat A(g)=g(La)\) for all \(g\in X'\), which is possible since a Hilbert space is reflexive. The operator \(L\), so defined, is linear and continuous (its norm does not exceed \(C_\varepsilon\)), and for all \(n\geq 0\), since the functionals \(\delta_n\) (derivatives of the delta-function) lie in \(E_0\), for the function \(\varphi(t)=(La)(t)\) we have
\[ \varphi^{(n)}(0)=\delta_n(\varphi)=\hat A(\delta_n)=A(\delta_n)=(-i)^n i^n a_n=a_n. \]
Thus the constructed operator \(L\) satisfies all the conditions of the theorem.

The constant \(\Gamma_\beta=(\cos \pi/2\beta)^{-\beta}\) occurring in the formulation of Theorem 1 is sharp in the sense that the following holds:

Theorem 1a. For any \(\varepsilon>0\) there exists a sequence
\((\xi_n)\in K(B^n n^\beta)\) for which there is no function \(\varphi(t)\) in the space
\(C^{\beta,\Gamma_\beta(B-\varepsilon)}\) such that
\(\varphi^{(n)}(0)=\xi_n,\ n\geq 0\).

Indeed, assuming the contrary, one can arrive at the assertion of Lemma 1, even with a more precise estimate of the coefficients of the function \(\Phi(s)\):
\[ |f_n|\leq \frac{A_\delta(1+\delta)^n\|f\|} {(B-\varepsilon)^n (n\cos \pi/2\beta)^{\beta n}}, \quad \delta>0, \]
and from these estimates—to the conclusion that every entire function of order \(F(s)\) with type \(B_1=B^{1/\beta}\) on the real axis has, in the whole plane, type not exceeding
\((B-\varepsilon)^{1/\beta}(\cos \pi/2\beta)^{-1}\). Examples of concrete functions (cf. \((^8)\), pp. 70–72) show that this assertion is false. Such is the scheme of the proof of Theorem 1a.

Lemma 1 and Theorem 1 are also valid in the multidimensional case. In what follows, for simplicity of formulation, we restrict ourselves to the two-dimensional case.

Let \(K(m_{nl})\) be the space of doubly indexed sequences
\[ \xi=(\xi_{nl}, n,l\geq 0) \]
with norm
\[ \|\xi\|=\sup_{n,l}\frac{|\xi_{nl}|}{m_{nl}}. \]

Theorem 1b. For arbitrary \(B\) and \(\varepsilon>0\) there exists a linear continuous operator
\[ L:K(B^{n+l}n^\beta l^\beta)\to C\{[(1+\varepsilon)\Gamma_\beta B]^{n+l}n^\beta l^\beta\} = C_2^{\beta,\Gamma_\beta B(1+\varepsilon)}, \]
such that
\[ D_t^nD_s^l(L\xi)\big|_{0,0}=\xi_{nl}. \]

1. Now, after a linear operator has been constructed for extending functions in the classes \(C^\beta\) from a point, one can extend functions from closed sets according to Whitney’s scheme \((^9)\) (see also \((^{10})\), § 3), with some refinements.

Theorem 2. Let \(F\) be an arbitrary closed set lying inside the unit square. One can construct a linear continuous operator \(L\) from \(C_2^{\beta B}\) to
\(C_2^{\beta,\Gamma_\beta(B+\varepsilon)}\) such that, for \(\psi=L\varphi\),
\[ D_t^nD_s^l\psi=D_t^nD_s^l\varphi \]
on \(F\) for all \(n,l\geq 0\), and
\[ \sup_{\substack{|t|,|s|\leq 1\\ n,l}} \frac{|D_t^nD_s^l\psi(t,s)|} {(\Gamma_\beta(B+\varepsilon))^{n+l}n^\beta l^\beta} \leq K_1 \sup_{\substack{(t,s)\in F\\ n,l}} \frac{|D_t^nD_s^l\varphi(t,s)|} {B^{n+l}n^\beta l^\beta}. \]

From Theorem 2 and the results of \((^{11})\) (Theorem 4) it follows*:

Theorem 3. A linear functional \(g\) on \(C_2^\beta\), if it is concentrated on a closed set \(F\) lying inside the unit square, admits the representation
\[ g(\varphi)=\sum_{n,l}\int_F D_t^nD_s^l\varphi(t,s)\,d\mu_{nl}, \]
where \(\mu_{nl}\) are measures on \(F\), and for every \(B\)
\[ \sum_{nl}\operatorname{Var}\mu_{nl}\cdot B^{n+l} n^{n\beta}l^{l\beta}<\infty . \]

Theorem 2 also makes it possible to extend to the case of spaces \(S_\alpha^\beta\) (see \((^4)\), pp. 210 and 282) Hörmander’s results \(((^{10}), \S 4, 5)\) on division by a polynomial.

1. For sequences \(m_n,\ n\geqslant0\), of a more general nature than \(n^{n\beta}\), \(\beta>1\), one can prove the following proposition:

Theorem 4. Let a sequence of positive numbers \(m_n,\ n\geqslant0\), satisfy the following conditions: 1) the ratios \(\mu_n=m_n/m_{n-1}\), \(\mu_0=1\), tend monotonically to infinity; 2) the series \(\sum \mu_n^{-1}\) converges, or, equivalently,
\[ \int_1^\infty \frac{\mu(t)}{t^2}\,dt<\infty, \]
where \(\mu(t)=\sup\{n:\mu_n\leqslant t\}\). Let the sequence \(p_n,\ n\geqslant0\), be defined by the relations
\[ p_n^{-1}=\inf\left\{r^{-n}\exp\left[r\int_1^\infty \frac{\mu(t)\,dt}{t(r+Dt)}\right]\right\}, \]
where \(D<1\).

Then there exists a linear continuous operator \(L\) from the Banach space of sequences \(K(p_n)\) into the Banach space of infinitely differentiable functions \(C(m_n)\) such that \((L\xi)^{(n)}(0)=\xi_n\) for all \(n\geqslant0\) and \(\xi\in K\).

The proof of this theorem is essentially the same as that of Theorem 1; in obtaining the analogue of Lemma 1, some estimates from \((^7)\) are used.

In conclusion—a few words on the functional dimension \(df\) (for the definition see \((^{12})\), p. 22) of the spaces \(S_\alpha^\beta\).

Theorem 5. The functional dimension
\[ df\{S^\beta[-1,1]\}=\beta . \]
In the case of several variables
\[ df\{S^{\beta_1\beta_2\ldots\beta_n}[-1,1]^n\} =\sum_{i=1}^n \beta_i,\quad df\{S_{\alpha_1\alpha_2\ldots\alpha_n}^{\beta_1\beta_2\ldots\beta_n}\} \]
\[ =\sum_{i=1}^n(\alpha_i+\beta_i). \]

Moscow State University
named after M. V. Lomonosov

Received
16 XII 1960

REFERENCES

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12. A. N. Kolmogorov, V. M. Tikhomirov, UMN, 14, issue 2, 3 (1959).

* In an imprecise form and without proof this fact was noted in \((^{11})\), Theorem 5.

** This space is defined analogously to the space \(C^{\beta B}\): the sequence \(B^n n^{n\beta}\) is replaced by \(m_n\).