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MATHEMATICS
B. I. KORENBLUM
THE WEIERSTRASS THEOREM IN SPACES OF INFINITELY DIFFERENTIABLE FUNCTIONS
(Presented by Academician A. N. Kolmogorov, 14 I 1963)
1°. In a report at the Fourth All-Union Mathematical Congress (Leningrad, 1961), the author of the present note showed that certain special rings of infinitely differentiable functions possess a peculiar structure of the set of primary ideals, forming transfinite chains. From the results of G. E. Shilov \((^1)\) it follows that in these rings even a local theorem of Weierstrass type on approximation by polynomials cannot hold. In connection with this there arises the question of the validity of a similar theorem in arbitrary Banach spaces of infinitely differentiable functions with a norm of type (1) (see below). In the present note a negative answer to this question is given. The structure of the corresponding subspaces in which the Weierstrass theorem holds is also clarified.
2°. Let \(\{A_n\}_0^\infty\) be a fixed increasing sequence of positive numbers, \(A_0=1\). We introduce the following Banach spaces:
a) \(D_{\{A_n\}}\) is the space of infinitely differentiable complex-valued functions \(f(x)\) \((-\infty < x < \infty)\) with norm
\[ \|f\|=\sup_{\substack{-\infty<x<\infty\\ n\geq 0}} \frac{|f^{(n)}(x)|}{A_n}<\infty; \tag{1} \]
b) \(\widetilde D_{\{A_n\}}\) is the subspace of \(D_{\{A_n\}}\) consisting of functions \(f(x)\) periodic with period \(2\pi\).
On the basis of well-known inequalities of A. N. Kolmogorov \((^2)\), relating the upper bounds of successive derivatives, the sequence \(\{A_n\}\) may be considered logarithmically convex, i.e.
\[ \frac{A_1}{A_0}\leq \frac{A_2}{A_1}\leq \frac{A_3}{A_2}\leq \cdots \tag{2} \]
We shall also assume that
\[ \lim_{n\to\infty}\frac{A_n}{A_{n-1}} = \lim_{n\to\infty}\sqrt[n]{A_n} =\infty; \tag{3} \]
otherwise, as is easy to see, \(\widetilde D_{\{A_n\}}\) is finite-dimensional and consists of trigonometric polynomials of bounded degree, while \(D_{\{A_n\}}\) consists of functions of exponential type with bounded exponent.
Denote by \(B\) the set of all entire functions \(p(x)\) of exponential type, bounded on the axis \(-\infty < x < \infty\), and by \(\widetilde B_n\) the set of all trigonometric polynomials
\[ \widetilde p(x)=\sum_{k=-n}^{n} c_k e^{ikx} \]
of arbitrary,
degree \(n\). By the classical inequality of S. N. Bernstein and (3), \(B \subset D\), \(\widetilde B \subset \widetilde D\).* Obviously, \(\widetilde B = B \cap \widetilde D\). Let \(D^0, \widetilde D^0\) denote, respectively, the closures of \(B\) and \(\widetilde B\) in \(D\).
Lemma.
\[ \widetilde D^0 = D^0 \cap \widetilde D . \tag{4} \]
Proof. Obviously, \(\widetilde D^0 \subset D^0 \cap \widetilde D\). We shall prove that \(D^0 \cap \widetilde D \subset \widetilde D^0\), i.e., that every periodic (with period \(2\pi\)) function from \(D\) which can be approximated with arbitrary accuracy (in the norm of \(D\)) by functions from \(B\) is also approximated by trigonometric polynomials. Let \(f(x)\in \widetilde D\), \(p(x)\in B\), \(\|f-p\|<\varepsilon\). Consider the functions
\[ p_n(x)=\frac{1}{n+1}\sum_{k=0}^{n} p(x+2k\pi)\quad (n=0,1,2,\ldots). \]
By the periodicity of \(f(x)\) we have
\[ \|f-p_n\|<\varepsilon\quad (n=0,1,2,\ldots). \]
On the other hand, using the uniform boundedness and equicontinuity on \((-\infty,\infty)\) of the family of functions \(\{p_n(x)\}\), as well as \(\{p_n^{(m)}(x)\}\) (for any fixed \(m\)), one can extract a subsequence \(\{p_{n_i}(x)\}\) converging, together with all its derivatives, to some function \(\widetilde p(x)\). Obviously, \(\widetilde p(x)\in B\) and \(\|f-\widetilde p\|\le \varepsilon\). Moreover, \(\widetilde p(x)\) is periodic, since
\[ p_n(x+2\pi)-p_n(x)=\frac{1}{n+1}\{p[x+2(n+1)\pi]-p(x)\}\to 0\quad (n\to\infty). \]
Therefore \(\widetilde p(x)\) is a trigonometric polynomial, as was required to prove.
3°. We proceed to the formulation of the main results.
Theorem 1. \(\widetilde D^0\) is a proper subspace of \(\widetilde D\); in other words, not every element \(f\in \widetilde D\) can be approximated with arbitrary accuracy by a trigonometric polynomial. Similarly, \(D^0\) is a proper subspace of \(D\).
Theorem 2. The space \(\widetilde D\) is nonseparable.
Theorem 3. In order that an element \(f\in D\) belong to \(D^0\), it is necessary and sufficient that either of the two equivalent conditions hold:
\[ 1)\quad \sup_{-\infty<x<\infty}|f^{(n)}(x)|=o(A_n)\quad (n\to\infty); \tag{5} \]
2) the translation operation \(f_\tau(x)=f(x-\tau)\) is strongly continuous in \(D\), i.e.
\[ \|f_\tau-f\|\to 0\quad (\tau\to 0). \tag{6} \]
These same conditions are necessary and sufficient in order that an element \(f\in \widetilde D\) belong to \(\widetilde D^0\).
4°. We first prove Theorem 3. Let \(p(x)\) be any function of the class \(B\). On the basis of S. N. Bernstein’s inequality, \(|p^{(n)}(x)|\le Ca^n\) \((-\infty<x<\infty;\ n=0,1,2,\ldots)\), where \(C,a\) are positive constants. Taking (3) into account, we obtain from this that \(p(x)\) satisfies (5). Since the set of elements \(f\in D\) for which (5) holds is closed, the necessity of condition (5) is proved. We shall show that (5) implies (6). Suppose that for some \(f\in D\) condition (5) is fulfilled and let \(\varepsilon>0\). There is an \(N\) such that
\[ |f^{(n)}(x)|<\frac{\varepsilon}{2}A_n\quad (-\infty<x<\infty) \]
for \(n>N\). There exists, further, a \(\delta\) such that for \(|\tau|<\delta\)
\[ |f^{(n)}(x+\tau)-f^{(n)}(x)|<\varepsilon A_n\quad (-\infty<x<\infty;\ n=0,1,\ldots,N). \]
Hence it follows that \(\|f_\tau-f\|<\varepsilon\) \((|\tau|<\delta)\), i.e. (6).
Now let \(f\in D\) satisfy condition (6); we shall prove that \(f\in D^0\).
* Here and in what follows we omit the index \(\{A_n\}\) for convenience.
Consider the following functions, which obviously belong to the class \(B\):
\[ P_n(x)=\frac{1}{\pi n}\int_{-\infty}^{\infty}\frac{\sin^2 n(x-\tau)}{(x-\tau)^2}f(\tau)\,d\tau =\frac{1}{\pi n}\int_{-\infty}^{\infty} f(x-\tau)\frac{\sin^2 n\tau}{\tau^2}\,d\tau = \]
\[ =\frac{1}{\pi n}\int_{-\infty}^{\infty} f_\tau \frac{\sin^2 n\tau}{\tau^2}\,d\tau \quad (n=1,2,\ldots), \tag{7} \]
where the last integral is understood as an abstract one in the space \(D\). We have
\[ \|P_n-f\|\leq \frac{1}{\pi n}\int_{-\infty}^{\infty}\|f_\tau-f\|\frac{\sin^2 n\tau}{\tau^2}\,d\tau =\frac{1}{\pi n}\left(\int_{-\infty}^{-\delta}+\int_{-\delta}^{\delta}+\int_{\delta}^{\infty}\right) \|f_\tau-f\|\frac{\sin^2 n\tau}{\tau^2}\,d\tau \leq \]
\[ \leq \frac{2\|f\|}{\pi n}\left(\int_{-\infty}^{-\delta}+\int_{\delta}^{\infty}\right) \frac{\sin^2 n\tau}{\tau^2}\,d\tau +\max_{|\tau|\leq\delta}\|f_\tau-f\| \leq \frac{4\|f\|}{\pi n\delta}+\max_{|\tau|\leq\delta}\|f_\tau-f\|. \]
Using (6), it is easy to obtain from the last estimate that \(\|P_n-f\|\to 0\) \((n\to\infty)\). The last assertion of Theorem 3 follows from what has just been proved and from (4).
We pass to the proof of Theorem 1. Denote \(A_{n-1}/A_n=\mu_n\) \((n=1,2,\ldots)\). By virtue of (2) and (3), \(\mu_n\) tends monotonically to zero \((n\to\infty)\). Consider the functions
\[ \chi_0(x)\equiv 1;\qquad \chi_n(x)=A_n\mu_n^n e^{ix/\mu_n} =\frac{\mu_n^n}{\mu_1\mu_2\cdots\mu_n}e^{ix/\mu_n} \quad (n=1,2,\ldots). \tag{8} \]
Let us note the following properties of \(\chi_n\):
a) \(\left|\chi_n^{(m)}(x)\right|\) does not depend on \(x\) \((n,m=0,1,2,\ldots)\);
b) \(\left|\chi_n^{(m)}(x)\right|\leq A_m\) \((n,m=0,1,2,\ldots)\);
c) \(\left|\chi_n^{(n)}(x)\right|=A_n\) \((n=0,1,2,\ldots)\);
d) \(\left|\chi_n^{(m)}(x)\right|\to 0\) \((n\to\infty,\ m\ \text{fixed})\);
e) \(\left|\chi_n^{(m)}(x)\right|=o(A_m)\) \((m\to\infty,\ n\ \text{fixed})\).
Using these properties, one can construct a sequence of integers
\[ n_0=N_0=0<n_1<N_1<n_2<N_2<\cdots<N_{k-1}<n_k<N_k<\cdots \]
so that the following conditions are satisfied: for all \(n\), \(N_{k-1}\leq n<N_k\),
\[ \left|\chi_{n_i}^{(n)}(x)\right|\leq \frac{A_n}{3^k}\quad (i<k); \tag{9} \]
\[ \left|\chi_{n_i}^{(n)}(x)\right|\leq \frac{A_n}{4^{\,i-k}}\quad (i>k). \tag{10} \]
Indeed, suppose \(n_1,N_1,n_2,N_2,\ldots,n_{k-1},N_{k-1}\) have already been chosen; using property e), choose \(N_k\) so that for \(n\geq N_{k-1}\) (9) is fulfilled; after this, using d), choose \(n_k\) so that for \(N_{i-1}\leq n<N_i\) \((i<k)\) the inequalities \(\left|\chi_{n_k}^{(n)}(x)\right|\leq A_n/4^{k-i}\) are fulfilled. We now show that the function
\[ \sigma(x)=\chi_{n_1}(x)+\chi_{n_2}(x)+\cdots \tag{11} \]
belongs to \(D\), but does not belong to \(D^0\). Let \(N_{k-1}\leq n<N_k\), and let \(x\) be arbitrary. By virtue of (9) and (10), as well as properties b) and c), we have
\[ \left|\sigma^{(n)}(x)\right| \leq \sum_{i=1}^{k-1}\left|\chi_{n_i}^{(n)}(x)\right| +\left|\chi_{n_k}^{(n)}(x)\right| +\sum_{i=k+1}^{\infty}\left|\chi_{n_i}^{(n)}(x)\right| \leq \]
\[ \leq \frac{A_n}{3}+A_n+A_n\sum_{i=1}^{\infty}4^{-i} =\frac{5}{3}A_n; \]
\[ |\sigma^{(n_k)}(x)|\ge |\chi_{n_i}^{(n_k)}(x)|-\sum_{i=1}^{k-1}|\chi_{n_i}^{(n_k)}(x)|-\sum_{i=k+1}^{\infty}|\chi_{n_i}^{(n_k)}(x)|\ge \]
\[ \ge A_{n_k}-\frac{A_{n_k}}{3}-A_{n_k}\sum_{i=1}^{\infty}4^{-i}=\frac{1}{3}A_{n_k}. \]
Therefore, at every point \(x\),
\[ |\sigma^{(n)}(x)|\le \frac{5}{3}A_n\quad (n=0,1,2,\ldots);\qquad \varlimsup_{n\to\infty}\frac{|\sigma^{(n)}(x)|}{A_n}\ge \frac{1}{3}. \tag{12} \]
On the basis of Theorem 3, \(\sigma\in D\), but \(\bar\sigma\notin D\). By modifying the construction somewhat, one can arrange that the functions \(\chi_{n_i}(x)\) are periodic with period \(2\pi\) and still satisfy (9) and (10). Then \(\sigma\in \widetilde D\), but \(\bar\sigma\notin \widetilde D^0\). Theorem 1 is proved.
Selecting from the sequence \(\{n_k\}\) all possible subsequences \(\{n_{k_\nu}\}\) and constructing the functions
\[ \sigma_{\{k_\nu\}}(x)=\chi_{n_{k_1}}(x)+\chi_{n_{k_2}}(x)+\cdots, \tag{13} \]
one can, by repeating the preceding arguments, show that for two different subsequences \(\{k_\nu\}\), \(\{k'_\nu\}\),
\(\|\sigma_{\{k_\nu\}}-\sigma_{\{k'_\nu\}}\|\ge 1/3\), provided only that the set
\((\{k_\nu\}\cup\{k'_\nu\})\setminus(\{k_\nu\}\cap\{k'_\nu\})\) is infinite. Thus there exists a continuum of elements \(\sigma_{\{k_\nu\}}\in \widetilde D\) whose mutual distances are not less than \(1/3\), which proves Theorem 2.
5°. To formulate the “local” Weierstrass theorem, introduce the concept of a norm on an interval \((a,b)\):
\[ \|f\|_{(a,b)}=\sup_{\substack{a<x<b\\ n=0,1,2,\ldots}}\frac{|f^{(n)}(x)|}{A_n}. \tag{14} \]
Theorem 4. In order that a function \(f(x)\in D\) be “locally approximable” by functions \(p(x)\in B\), i.e., that for every finite interval \((a,b)\) and every \(\varepsilon>0\) there exist a function \(p(x)\in B\) such that \(\|f-p\|_{(a-b)}<\varepsilon\), it is necessary and sufficient that
\[ \sup_{a<x<b}|f^{(n)}(x)|=o(A_n)\qquad (n\to\infty), \tag{15} \]
whatever the finite interval \((a,b)\) may be.
To prove Theorem 4 it is necessary to consider the functions \(P_n(x)\) (see (7)) and to refine somewhat the argument used in the proof of Theorem 3.
Theorem 4 remains valid if, in it, the class \(B\) is replaced by the set of finite trigonometric sums \(\sum c_n e^{i\gamma_n x}\) (\(c_n\) arbitrary complex numbers, \(\gamma_n\) arbitrary real numbers). To prove this, one must suitably approximate the functions (7) by trigonometric sums, for example by B. M. Levitan’s polynomials \((^3)\).
Let us note that, by virtue of (12), the function \(\sigma(x)\) cannot be approximated on any interval by functions of the class \(B\).
Kyiv Civil Engineering Institute
Received
10 I 1963
REFERENCES
- G. E. Shilov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 21 (1947).
- A. N. Kolmogorov, Uch. zap. Mosk. univ., 30, 3 (1939).
- B. M. Levitan, DAN, 15, 169 (1937).