UDC 517.11

MATHEMATICS

FAN DINH ZIEU

ON SOME PROPERTIES OF CONSTRUCTIVE GENERALIZED FUNCTIONS

(Presented by Academician P. S. Novikov, October 22, 1966)

1. The present note is a continuation of the author’s work (⁵). In this note all special terms and notations not explained are understood in the same way as in (²–⁵). In (⁵) the notion of a constructive generalized function was defined; it is a constructive analogue of the classical notion of a generalized function defined in (⁶, ⁷).

Let us introduce some further notions and notations: let \(\Phi_1,\Phi_2\) be constructive generalized functions and let \(a,b\) be real duplexes, \(a<b\). Take a fixed natural number \(k\) such that \(k \geq \max(|a| \square |b|)\). We shall say that \(\Phi_1\) is equal to \(\Phi_2\) in the interval \(a \nabla b\), if there exists a word \(P\) of type ю such that \(L(P) < l^*\) and

\[ \forall x(a<x<b \supset I(k \square l-m \square \{\Phi_1\}_k)(x)- \]

\[ -I(k \square l-n \square \{\Phi_2\}_k)(x)=G(P)(x)), \]

where \(m \rightleftarrows |\Phi_1|_k,\ n \rightleftarrows |\Phi_2|_k,\ l \rightleftarrows \max(m \square n)\), and \(G\) is the algorithm defined in Sec. 1 of (⁵).

It is not difficult to prove that in the indicated definition, instead of \(k\) one may take any natural number greater than or equal to \(\max(|a| \square |b|)\).

We shall say that \(\Phi_1\) is equal to \(\Phi_2\) in the interval \(a \nabla \infty\), if there exists an algorithm \(\mathfrak P\) of type \((\mathrm{н}\dot{\to}\mathrm{ю})\) possessing the following properties:

1) \(\forall k\ (k>a \supset !\mathfrak P(k));\)

2) \(\forall k\ (k>a \supset L(\mathfrak P(k))<l_k);\)

3) \(\forall kx\ (a<x\leq k \supset I(k \square l_k-m_k \square \{\Phi_1\}_k)(x)-\)

\[ -I(k \square l_k-n_k \square \{\Phi_2\}_k)(x)=G(\mathfrak P(k))(x)), \]

where \(m_k \rightleftarrows |\Phi_1|_k,\ n_k \rightleftarrows |\Phi_2|_k,\ l_k \rightleftarrows \max(m_k \square n_k)\).

In an analogous way one may define the notion that \(\Phi_1\) is equal to \(\Phi_2\) in the interval \(-\infty \nabla a\).

Theorem 1. There exists an algorithm that transforms every word of the form \(a \nabla b \square \Phi_1\), where \(a,b\) are real duplexes, \(a<b\), and \(\Phi_1\) is a constructive generalized function, into a word of the form \(n \square \mathfrak m_1\) such that \(\Phi_1\) is equal to \(\mathfrak D(n \square R(\mathfrak m_1))\) in the interval \(a \nabla b\).

In other words, in every finite interval every constructive generalized function is equal to a derivative of some order of an almost uniformly continuous function.

* The notation \(<\) is understood in the same way as in (⁵).

We shall say that the segment \(a \triangle b\) is a carrier of the constructive generalized function \(\Phi_1\), and shall write

\[ (a \triangle b \text{ carries } \Phi_1), \]

if \(\Phi_1\) is equal to zero in the intervals \(-\infty \nabla a\) and \(b \nabla \infty\).

Let \(\Phi_1\) be a constructive generalized function. We shall say that \(\Phi_1\) is a constructive generalized function with bounded carrier if

\[ \exists k\,(-k \triangle k \text{ carries } \Phi_1). \]

Theorem 2. There is no algorithm that transforms each constructive generalized function \(\Phi_1\) with bounded carrier into a natural number \(k\) such that \((-k \triangle k \text{ carries } \Phi_1)\).

Theorem 3. a) Every constructive generalized function with bounded carrier is a constructive generalized function of finite order; b) there is no algorithm that transforms each constructive generalized function with bounded carrier \(\Phi_1\) into a natural number \(n\) such that \(\Phi_1\) is a constructive generalized function of order \(\leq n\).

2. Passing to the consideration of sequences of constructive generalized functions and of the limits of such sequences, we first introduce some notation: let \(\mathfrak{m}_1\) be a word of type \(\mathfrak{m}\), i.e. the complete code of an almost uniformly continuous function. We shall denote \(\mathfrak{D}(n \square R(\mathfrak{m}_1))\) by \(\mathfrak{m}_1^{(n)}\). If two constructive generalized functions \(\Phi_1\) and \(\Phi_2\) are equal in the interval \(-k \nabla k\), we shall write \(\Phi_1 \underset{k}{=} \Phi_2\).

Let \(\lambda_1\) be an algorithm of type \((\mathfrak{n}\to\mathfrak{m})\) and \(X\) a word of type \(\mathfrak{m}\). We shall write \((X \,\mathrm{pred}_{C_k}\lambda_1)\) if \(\xi(k \square X)\) is the limit of the sequence defined by the algorithm \((\tilde{\xi}_k \square \circ \lambda_1)\) in the space \(C_k\).

Let \(\varphi\) be an algorithm of type \((\mathfrak{n}\to\Phi)\), i.e. a sequence of constructive generalized functions. We shall say that the constructive generalized function \(\Phi_1\) is the limit of the sequence \(\varphi\) in the interval \(-k \nabla k\) if

\[ \exists i\,\mathfrak{m}_1\lambda_1\bigl(\Phi_1 \underset{k}{=} \mathfrak{m}_1^{(i)} \ \&\ \forall_n^{\sim}(\varphi(n) \underset{k}{=} \lambda_1(n)^{(i)}) \ \&\ (\mathfrak{m}_1\,\mathrm{pred}_{C_k}\lambda_1)\bigr), \]

where \(\lambda_1\) is a variable for algorithms of type \((\mathfrak{n}\to\mathfrak{m})\).

We shall call \(\Phi_1\) the limit of the sequence \(\varphi\), if for every \(k\), \(\Phi_1\) is the limit of the sequence \(\varphi\) in the interval \(-k \nabla k\).

Theorem 4. a) Every constructive generalized function is the limit of a sequence of regular constructive generalized functions; b) every constructive generalized function is the limit of a sequence of constructive generalized functions with bounded carriers.

It is not difficult to prove the following assertions:

1) If \(\mathfrak{m}_1\) is the limit in the space \(\mathcal{C}\) of a sequence \(\lambda\) of words of type \(\mathfrak{m}\) (i.e. for every \(k\), \(\xi(k \square \mathfrak{m}_1)\) is the limit in \(C_k\) of the sequence defined by the algorithm \((\tilde{\xi}_k \square \circ \lambda)\)), then \(R(\mathfrak{m}_1)\) is the limit of a sequence of constructive generalized functions defined by the algorithm \((R\circ\lambda)\).

2) If \(\Phi_1\) is the limit of the sequence \(\varphi\) of constructive generalized functions, then for whatever natural number \(m\), \(\mathfrak{D}(m \square \Phi_1)\) is the limit of the sequence of constructive generalized functions defined by the algorithm \((\mathfrak{D}_m \square \circ \varphi)\).

Let a real duplex \(a_0\) be given. Let \(k\) be a natural number, \(\mathfrak{m}_1^k\) a word of type \(\mathfrak{m}^k\), and \(\lambda\) an algorithm of type \((\mathfrak{d}\dot{\to}\mathfrak{m}^k)\), applicable to any-

to any duplex \(\alpha \ne \alpha_0\). We shall say that \(\mathfrak m_1^k\) is the limit of \(\lambda\) as \(\alpha \to \alpha_0\) in the space \(C_k\), if

\[ \forall i\, \exists j\, \forall \alpha\, \bigl(-k \le x \le k \,\&\, 0<|\alpha-\alpha_0|<2^{-j} \supset |\mathfrak m_1^k(x)-\lambda(\alpha)(x)|<2^{-i}\bigr). \]

Let \(\mathfrak m_1\) be a word of type \(\mathfrak m\), and let \(\lambda\) be an algorithm of type \((\mathfrak d \to \mathfrak m)\), applicable to every duplex \(\alpha \ne \alpha_0\). We shall say that \(\mathfrak m_1\) is the limit of \(\lambda\) as \(\alpha \to \alpha_0\) in the space \(C_k\), and shall write

\[ (\mathfrak m_1 \operatorname{pred}_{C_k(\alpha\to\alpha_0)} \lambda), \]

if \(\xi(k \square \mathfrak m_1)\) is the limit of \((\widetilde{\xi}_{k\square}\circ\lambda)\) as \(\alpha \to \alpha_0\) in the space \(C_k\).

Now let \(\varphi\) be an algorithm of type \((\mathfrak d \to \Phi)\), applicable to every duplex \(\alpha \ne \alpha_0\). We shall say that the constructive generalized function \(\Phi_1\) is the limit of \(\varphi\) as \(\alpha \to \alpha_0\), if for every natural number \(k\) the following holds:

\[ \exists i j \mathfrak m_1 \lambda_1 \bigl(\Phi_1=\mathfrak m_1^{(i)} \,\&\, \forall \alpha\,(0<|\alpha-\alpha_0|<2^{-j}\supset \]

\[ \supset \varphi(\alpha)=\lambda_1(\alpha)^{(i)} \,\&\, (\mathfrak m_1 \operatorname{pred}_{C_k(\alpha\to\alpha_0)} \lambda_1)\bigr), \]

where \(\lambda_1\) is a variable for algorithms of type \((\mathfrak d \to \mathfrak m)\), applicable to any duplex \(\alpha \ne \alpha_0\).

3. We define the operator of a linear change of variable for constructive generalized functions.

Let \(\alpha \ne 0\) and \(\beta\) be real duplexes, and let \(\Phi_1 \simeq X\tau Y\tau Z\tau T\) be a constructive generalized function. Construct an algorithm \(\eta\) of type \((\mathfrak n \to \mathfrak n)\) such that

\[ \forall k\,(\eta(k)>|\alpha|k+|\beta|). \]

Construct algorithms \(\lambda\) of type \((\mathfrak n \to \mathfrak m)\), \(N\) of type \((\mathfrak n \to \mathfrak n)\), \(Q\) of type \((\mathfrak n\mathfrak n \to \mathfrak n)\), such that

\[ \forall n x\,(\lambda(n)(x) \simeq \langle X\rangle(n)(\alpha x+\beta)); \]

\[ \forall k\,(N(k) \simeq \langle Y\rangle(\eta(k))); \]

\[ \forall k m\,(Q(k\square m) \simeq \langle T\rangle(\eta(k)\square m)). \]

It is possible to construct an algorithm \(P\) of type \((\mathfrak n\mathfrak n \to \mathfrak y)\) such that, for any \(k\) and \(n\), \(L(P(k\square n))<p\) \((p \simeq N(k))\), and for every \(x\) in \(-k \triangle k\):

\[ \mathfrak S(p\square \lambda(n))(x)+G(P(k\square n))(x)= \]

\[ =\frac{1}{\alpha^p}\cdot \mathfrak S(p\square \langle X\rangle(n))+ G(\langle Z\rangle(\eta(k)\square n))(\alpha x+\beta). \]

It is not difficult to prove that the word \(\xi\lambda3\tau\xi N3\tau\xi P3\tau\xi Q3\) represents a constructive generalized function. One can construct an algorithm which transforms each word of the form \(\alpha\square\beta\square\Phi_1\) into the corresponding word \(\xi\lambda3\tau\xi N3\tau\xi P3\tau\xi Q3\). We shall denote this algorithm by \(\mathcal L\).

Let \(c\) be a real duplex. Denote by \(\widetilde c\) a word of type \(\mathfrak m\) such that

\[ \forall x\,(\widetilde c(x)=c), \]

and then denote \(R(\widetilde c)\) by \(\bar c\). Then \(\bar c\) is a constructive generalized function.

A constructive generalized function \(\Phi_1\) is called constant if there exists a real duplex \(c\) such that \(\Phi_1=\bar c\).

Let a constructive generalized function \(\Phi_1\) and a real duplex \(x_0\) be given. We construct an algorithm \(\varphi\) of type \((\mathrm{D}\dot{\to}\Phi)\), such that for any \(a\ne 0\):

\[ \varphi(a) \simeq \mathscr{L}(a \square x_0 \square \Phi_1). \tag{1} \]

Copying the classical proof (see \((^6)\), § 16), we can prove the following assertion: if there exists such a constructive generalized function \(\Phi_2\) that \(\Phi_2\) is the limit of \(\varphi\) as \(a\to 0\), then \(\Phi_2\) is constant.

We shall say that a real duplex \(c\) is a value of the constructive generalized function \(\Phi_1\) at the point \(x_0\), and shall write

\[ (c\ \underline{\mathrm{val}}_{x_0}\ \Phi_1), \]

if \(\bar c\) is the limit of the algorithm \(\varphi\), defined by formula (1), as \(a\to 0\).

The point \(x_0\) is called a regular point of the constructive generalized function \(\Phi_1\) if \(\exists c(c\ \underline{\mathrm{val}}_{x_0}\ \Phi_1)\); it is called a singular point of \(\Phi_1\) if it is not a regular point of \(\Phi_1\).

It is easy to prove that if \(\Phi_1\) is equal to \(R(\mathfrak{sh}_1)\) in some interval \(a \nabla b\), where \(\mathfrak{sh}_1\) is a word of type \(\mathfrak{sh}\), then every point \(x_0\) in \(a \nabla b\) is a regular point of \(\Phi_1\), and whatever the point \(x_0\) in \(a \nabla b\), \((\mathfrak{sh}_1(x_0)\underline{\mathrm{val}}_{x_0}\Phi_1)\).

Łojasiewicz’s theorem (Theorem 2.3 \((^8)\)) is transferred into constructive mathematics in the following way.

Let \(\Phi_1\) be a constructive generalized function and \(x_0\) a real duplex. Let \(k>|x_0|\). The real duplex \(c\) is a value of \(\Phi_1\) at the point \(x_0\) if and only if there exist a number \(n\) and a word \(\mathfrak{sh}_1\) such that \(\Phi_1\) is equal to \(\mathfrak{sh}_1^{(n)}\) in \(-k \nabla k\), and \(c/n!\) is the limit of the function \(f\), defined by the formula

\[ \forall x\,(f(x)\simeq \mathfrak{sh}_1(x):(x-x_0)^n), \]

as \(x\to x_0\).

Theorem 5. One can construct a constructive generalized function \(\Phi_1\) possessing the property: there is no function \(f\) of a real variable, defined at all regular points of \(\Phi_1\), and such that at every regular point \(x_0\) of \(\Phi_1\) one has \((f(x_0)\underline{\mathrm{val}}_{x_0}\Phi_1)\).

The proof of this theorem is based on Theorem 5.1 from \((^1)\) and the lemma of § 1, Chapter III from \((^4)\).

Theorem 6. There exists such a constructive everywhere-defined function \(f\) of a real variable that there is no constructive generalized function \(\Phi_1\) satisfying the condition:

\[ \forall x\,(f(x)\underline{\mathrm{val}}_x\Phi_1). \]

The question of a constructive analogue of the concept of generalized function, based on the concept of a linear continuous functional in the space of finite infinitely differentiable functions, will be considered in another communication.

The author expresses deep gratitude to A. A. Markov and N. A. Shanin for valuable advice.

Moscow State University
named after M. V. Lomonosov

Received
15 X 1966

References

\({}^1\) A. A. Markov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 52, 315 (1958).
\({}^2\) N. A. Shanin, ibid., 52, 226 (1958).
\({}^3\) N. A. Shanin, ibid., 67, 15 (1962).
\({}^4\) G. S. Tseitin, ibid., 67, 295 (1962).
\({}^5\) Fan Ding Zieu, DAN, 174, No. 1 (1967).
\({}^6\) J. Mikusinski, R. Sikorski, The Elementary Theory of Distributions, Warszawa, 1, 1957; 2, 1959.
\({}^7\) J. Korevaar, Indagation Math., 17, 3, 368; 17, 4, 483; 17, 663 (1955).
\({}^8\) S. Łojasiewicz, Studia Math., 16, 1, 1 (1957).