UDC 517.11

MATHEMATICS

FAN DINH ZIEU

CONSTRUCTIVE GENERALIZED FUNCTIONS

(Presented by Academician P. S. Novikov on 22 X 1966)

1. The present note is devoted to defining the concept of a generalized function in constructive mathematics. First of all we introduce some concepts and notation*.

By a word of type \(\mathfrak{ш}^{k}\) we shall mean the complete code of a uniformly continuous function on the interval \(-k\Delta k\), where \(k\) is an arbitrary natural number greater than 0. By \(C_k\) we shall denote the normed space of words of type \(\mathfrak{ш}^{k}\). Let us note that if \(k \leq l\), then every word of type \(\mathfrak{ш}^{l}\) is a word of type \(\mathfrak{ш}^{k}\). The equality relation, the addition algorithm, and the algorithm of multiplication by real duplexes in the space \(C_k\) will be denoted respectively by
\[ \underset{\mathfrak{ш},k}{=},\quad \underset{\mathfrak{ш},k}{+},\quad \underset{\mathfrak{ш},k}{\cdot}. \]

Let \(f\) be an everywhere defined constructive function. We shall say that \(f\) is an almost uniformly continuous function if there is an algorithm \(g\) of type \((\mathfrak{н}\to\mathfrak{н})\) such that, for every \(k \geq 1\), \(g_k\) is a regulator of uniform continuity of the function \(f\) on the interval \(-k\Delta k\).

The algorithm \(g\) is called a regulator of almost uniform continuity of the function \(f\). The word \(\xi f\mathfrak{з}\overline{1}\xi g\mathfrak{з}\) will be called the complete code of the almost uniformly continuous function, and also a word of type \(\mathfrak{ш}\). One can define an equality relation \(\mathfrak{ш}\), an addition algorithm \(+\), and an algorithm of multiplication by real duplexes \(\cdot\) in the usual way, so that the set of words of type \(\mathfrak{ш}\) forms a constructive linear space. We denote this space by \(\mathcal{C}\). It is not difficult to construct an algorithm \(\xi\) having the following property: for every \(k \geq 1\), \(\tilde{\xi}_k\) transforms any word of the form \(\xi f\mathfrak{з}\overline{1}\xi g\mathfrak{з}\) into the word \(\xi f\mathfrak{з}\overline{1}\xi\tilde{g}_{k}\mathfrak{з}\). Obviously, \(\xi_k\) transforms every word of type \(\mathfrak{ш}\) into a word of type \(\mathfrak{ш}^{k}\). Let \(\mathfrak{ш}_1\) and \(\mathfrak{ш}_2\) be words of type \(\mathfrak{ш}\). Instead of the formula
\[ \xi(k\,\square\,\mathfrak{ш}_1) \underset{\mathfrak{ш},k}{=} \xi(k\,\square\,\mathfrak{ш}_2) \]
we shall simply write
\[ \mathfrak{ш}_1 \underset{\mathfrak{ш},k}{=} \mathfrak{ш}_2. \]

We can construct an algorithm \(J_1\), which, when applied to any word of the form \(k\,\square\,x\,\square\,y\,\square\,\mathfrak{ш}_1^{k}\), where \(k\) is a natural number \(\geq 1\), \(x\) and \(y\) are real duplexes, \(-k \leq x \leq y \leq k\), and \(\mathfrak{ш}_1^{k}\) is a word of type \(\mathfrak{ш}^{k}\), transforms every such word into the integral of the complete code \(\mathfrak{ш}_1^{k}\) in the space \(\mathfrak{L}_1(x,y)\) (see \((^2)\), § 15.5). Then we construct an algorithm \(J\) such that, for every \(k \geq 1\), \(J_k\) is an algorithm of type \((\mathfrak{ш}^{k}\to\mathfrak{ш}^{k})\) and such that, for any number \(k \geq 1\), any word \(\mathfrak{ш}_1^{k}\) of type \(\mathfrak{ш}^{k}\), and any duplex \(x\), \(-k \leq x \leq k\), the following holds:
\[ \underbrace{J(k\,\square\,\mathfrak{ш}_1^{k})}_{\mathfrak{ш},k}(x) \simeq J_1\bigl(k\,\square\,\min(x\,\square\,0)\,\square\,x\,\square\,\mathfrak{ш}_1^{k}\bigr) - J_1\bigl(k\,\square\,\min(x\,\square\,0)\,\square\,0\,\square\,\mathfrak{ш}_1^{k}\bigr). \]

* All specifically unexplained terms and notation are understood in the same way as in \((^{1-3})\).

The number \(J(k \Box \mathfrak{sh}_1^k)(x)\) is called the integral of the complete cipher \(\mathfrak{sh}_1^k\) from \(0\) to \(x\), and \(J(k \Box \mathfrak{sh}_1^k)\) is called the primitive of the complete cipher \(\mathfrak{sh}^k\) on the segment \(-k\Delta k\). Now we can construct an algorithm \(I\) such that, for every \(k \ge 1\), \(I_k\) is an algorithm of type \((\mathfrak{n}\mathfrak{sh}^k \to \mathfrak{sh}^k)\) and the following scheme is satisfied:

\[ I(k \Box 0 \Box \mathfrak{sh}_1^k) \simeq \mathfrak{sh}_1^k; \qquad I(k \Box n+1 \Box \mathfrak{sh}_1^k) \simeq J(k \Box I(k \Box n \Box \mathfrak{sh}_1^k)). \]

\(I(k \Box n \Box \mathfrak{sh}_1^k)\) is called the primitive of order \(n\) of the complete cipher \(\mathfrak{sh}_1^k\) on the segment \(-k\Delta k\).

After this one can construct an algorithm \(\mathfrak{I}\) of type \((\mathfrak{n}\mathfrak{sh}\to\mathfrak{sh})\), satisfying the following condition for any numbers \(k,n\) (\(k \ge 1\)) and any word \(\mathfrak{sh}_1\) of type \(\mathfrak{sh}\):

\[ \xi(k \Box \mathfrak{I}(n \Box \mathfrak{sh}_1)) = I(k \Box n \Box \xi(k \Box \mathfrak{sh}_1)). \]

\(\mathfrak{I}(n \Box \mathfrak{sh}_1)\) is called the primitive of order \(n\) of the complete cipher \(\mathfrak{sh}_1\).

We shall call every \(\sigma\)-system of real duplexes a polynomial germ, and also a word of type \(\mathfrak{yu}\) \((^2)\). Let \(L\) denote an algorithm computing the number of \(\sigma\)-terms in a word of type \(\mathfrak{yu}\), and let \(G\) denote an algorithm of type \((\mathfrak{yu}—\mathfrak{sh})\) having the following property: whatever the word \(\mathfrak{yu}_1\),

\[ \forall_x\bigl(G(\mathfrak{yu}_1)(x) \simeq a_1+a_2\cdot x+\cdots+a_n\cdot x^{\,n-1}\bigr), \]

where \(n \simeq L(\mathfrak{yu}_1)\), and \(a_i\) is the \(i\)-th \(\sigma\)-term of the word \(\mathfrak{yu}_1\) \((i=1,\ldots,n)\). We note that all judgments constructed with the aid of the indicated concepts and notation can be formulated rigorously within the framework of the logico-mathematical languages of N. A. Shanin \((^1)\).

2. We now define the concept of a constructive generalized function. In order to define this concept, we shall rely on the classical definition of Mikusinski—Sikorski—Korevaar \((^{7,8})\).

Of an algorithm \(\lambda\) of type \((\mathfrak{n}—\mathfrak{sh})\) we shall say that it is a fundamental sequence of words of type \(\mathfrak{sh}\) if potentially realizable algorithms \(N\) of type \((\mathfrak{n}\to\mathfrak{n})\), \(P\) of type \((\mathfrak{n}\mathfrak{n}\to\mathfrak{yu})\), and \(Q\) of type \((\mathfrak{n}\mathfrak{n}\to\mathfrak{n})\) exist, satisfying the following conditions:

\[ 1)\quad \forall kn\bigl(L(P(k \Box n)) < N(k)\bigr)^*; \]

\[ 2)\quad \forall k i j m x\bigl(|x|\le k\ \&\ i,j \ge Q(k \Box m)\supset |F(k \Box i)(x)-F(k \Box j)(x)|<\overline{2}^{\,m}\bigr). \]

where \(F\) is an algorithm of type \((\mathfrak{n}\mathfrak{n}\to\mathfrak{sh})\), defined as follows:

\[ \forall kn\bigl(F(k \Box n)\simeq J(N(k)\Box \lambda(n))+G(P(k \Box n))\bigr). \]

Of a word \(X\) in the alphabet \(\mathfrak{Ch}_0\) we shall say that it is an \(F\)-generalized function if the algorithm \(\langle X\rangle\) is a fundamental sequence of words of type \(\mathfrak{sh}\).

Of a word \(A\) in the alphabet \(\mathfrak{Ch}_0\cup\{t\}\) we shall say that it is an \(FR\)-generalized function if it has the form \(XtYtZtT\), where \(X\) is a record of some algorithm \(\lambda\) of type \((\mathfrak{n}\to\mathfrak{sh})\), \(Y\) is a record of some algorithm \(N\) of type \((\mathfrak{n}\to\mathfrak{n})\), \(Z\) is a record of some algorithm \(P\) of type \((\mathfrak{n}\mathfrak{n}\to\mathfrak{yu})\), \(T\) is a record of some algorithm \(Q\) of type \((\mathfrak{n}\mathfrak{n}\to\mathfrak{n})\), and conditions 1)—2) are satisfied.

An \(FR\)-generalized function we shall also call a constructive generalized function or a word of type \(\Phi\). It is easy to see that the set of constructive generalized functions can be defined by a normal formula.

\[ {}^*\ \text{The notation } < \text{ is understood in the following way:} \]

\[ \bigl(L(\mathfrak{yu}_1)<n\bigr) \Leftrightarrow \bigl(n\ge 1\supset L(\mathfrak{yu}_1\le n)\ \&\ (n=0\supset \mathfrak{yu}_1=0)\bigr), \]

where \(\mathfrak{yu}_1\) is a word of type \(\mathfrak{yu}\) and \(n\) is a natural number.

Let \(\Phi_1 \rightleftarrows \xi \lambda \mathcal E N\mathcal E P\mathcal E Q\mathcal E\) be a constructive generalized function. We agree to denote:
\[ |\Phi_1|_k \rightleftarrows N(k);\qquad [\Phi_1]_{k,n}\rightleftarrows P(k\square n). \]
According to condition 2), the algorithm \((\xi_k \circ \bar F_{k\square})\) determines a sequence of words of type \(\mathrm{ш}^k\) converging in itself, with convergence regulator \(Q_{k\square}\). One can construct an algorithm that transforms each word of the form \(k\square\Phi_1\) into the limit of the sequence \((\xi_{k\square}\circ\bar F_{k\square})\) in the space \(C_k\). We denote this limit by \(\{\Phi_1\}_k\). This is a word of type \(\mathrm{ш}^k\).

Let two constructive generalized functions \(\Phi_1\) and \(\Phi_2\) be given. Denote
\[ p_k \rightleftarrows |\Phi_1|_k,\qquad q_k \rightleftarrows |\Phi_2|_k,\qquad l_k \rightleftarrows \max(p_k\square q_k). \]
We shall say that \(\Phi_1\) is equal to \(\Phi_2\) and shall write \(\Phi_1=\Phi_2\), if there is an algorithm \(\mathcal P\) of type \((\mathrm{н}\to\mathrm{ю})\) satisfying the conditions:
\[ \begin{aligned} 1)\;& \forall k\bigl(\bar L(\mathcal P(k))<l_k\bigr);\\ 2)\;& \forall k\bigl(I(k\square l_k-p_k\square\{\Phi_1\})= I(k\square l_k-q_k\square\{\Phi_2\}_k)+\xi(k\square G(\mathcal P(k)))\bigr). \end{aligned} \]

Theorem 1. There is an algorithm that transforms each pair of equal constructive generalized functions \(\Phi_1\) and \(\Phi_2\) into a notation of some algorithm \(\mathcal P\) of type \((\mathrm{н}\to\mathrm{ю})\) satisfying conditions 1)—2).

It follows from Theorem 1 that the relation of equality between constructive generalized functions can be defined by a normal formula.

Let \(\Phi_1 \rightleftarrows X\tau Y\tau Z\tau T\) be a constructive generalized function. The word \(X\) will be called the basis of the generalized function \(\Phi_1\). We shall say that the constructive generalized function \(\Phi_1\) is equal to the \(F\)-generalized function \(X\), if it is equal to some constructive generalized function whose basis is \(X\).

Theorem 2. a) For every \(F\)-generalized function one can construct an equal constructive generalized function; b) there is no algorithm that transforms every \(F\)-generalized function into an equal constructive generalized function.

It is not difficult to define addition of constructive generalized functions, multiplication of constructive generalized functions by real duplexes, and the zero constructive generalized function in such a way that the set of constructive generalized functions forms a linear space (see the definition in (6)).

1. We now define the derivative of a constructive generalized function.

Let \(\Phi_1 \rightleftarrows X\tau Y\tau Z\tau T\) be a constructive generalized function. Construct algorithms \(\gamma\) and \(\delta\) of type \((\mathrm{н}\to\mathrm{н})\), such that for any natural numbers \(k,n\):
\[ \gamma(n)\simeq \langle X\rangle(n)(n\square n);\qquad \delta(k)\simeq \mu j\left(2^j>\frac{4k^{|\Phi_1|_k}}{|\Phi_1|_{k-1}!}\right). \]

After this, construct algorithms \(\lambda\) of type \((\mathrm{н}\to\mathrm{ш})\), \(N\) of type \((\mathrm{н}\to\mathrm{н})\), \(P\) of type \((\mathrm{нн}\to\mathrm{ю})\), and \(Q\) of type \((\mathrm{нн}\to\mathrm{н})\), satisfying the following conditions for any natural numbers \(k,n,m\) and any duplex \(x\):
\[ \lambda(n)(x)\simeq 2^{\gamma(n)}\bigl(\langle X\rangle(n)(x+2^{-\gamma(n)})-\langle X\rangle(n)(x)\bigr);\qquad N(k)\simeq |\Phi_1|_k+1; \]
\[ P(k\square n)\simeq [\Phi_1]_{k,n} + \underbrace{0\sigma\ldots\sigma0\sigma}_{|\Phi_1|_k\ \text{times}} \frac{\langle X\rangle(n)(0)}{|\Phi_1|_k!}; \]
\[ Q(k\square m)\simeq \max\bigl(m+1+\delta(k)\square k+1\square \langle T\rangle(k\square m+1)\bigr). \]

Put
\[ X_1 \rightleftarrows \mathcal E\lambda\mathcal E,\qquad Y_1 \rightleftarrows \mathcal E N\mathcal E,\qquad Z_1 \rightleftarrows \mathcal E P\mathcal E,\qquad T_1 \rightleftarrows \mathcal E Q\mathcal E. \]
It is easy to prove that the word \(X_1\tau Y_1\tau Z_1\tau T_1\) is a constructive generalized function. We shall call this generalized function the derivative of the constructive generalized function \(\Phi_1\). One can construct an algorithm \(D\),

constructing, for each constructive generalized function, its derivative. After this one can construct an algorithm \(\mathfrak D\) of type \((\mathfrak{nf}\to \Phi)\) satisfying the scheme:
\[ \mathfrak D(n \square \Phi_1)\simeq \Phi_1;\qquad \mathfrak D(n+1 \square \Phi_1)\simeq D(\mathfrak D(n \square \Phi_1)). \]
\(\mathfrak D(n \square \Phi_1)\) is called the derivative of \(n\)-th order of \(\Phi_1\). Thus every constructive generalized function has derivatives of arbitrary order.

1. Denote by \(R\) an algorithm of type \((\mathfrak m\to \Phi)\) which transforms every word \(\mathfrak m_1\) of type \(\mathfrak m\) into a word \(XtYtZtT\) of type \(\Phi\) such that
   \[ \forall_n(\langle X\rangle(n)\doteq \mathfrak m_1);\qquad \forall_k(\langle Y\rangle(k)\doteq 0); \]
   \[ \forall_{kn}(\langle Z\rangle(k \square n)\doteq 0);\qquad \forall_{km}(\langle T\rangle(k \square m)\doteq 0). \]
   We shall say that \(R(\mathfrak m_1)\) is a representative of the complete cipher of an almost uniformly continuous function \(\mathfrak m_1\) in the space of constructive generalized functions.

Let \(\Phi_1\) be a constructive generalized function. We shall say that \(\Phi_1\) is a generalized function of order \(\leqslant n\), if
\[ \exists \Phi_2(\Phi_1=\Phi_2 \& \forall k(|\Phi_2|_k \geq n)); \]
that \(\Phi_1\) is a generalized function of finite order if there exists a number \(n\) such that \(\Phi_1\) is a generalized function of order \(\leqslant n\); that \(\Phi_1\) is a generalized function of \(n\)-th order if it is a generalized function of order \(\leqslant n\) and, when \(n\geq 1\), it is not a generalized function of order \(\leqslant n-1\); that \(\Phi_1\) is a regular generalized function if it is a generalized function of order \(0\).

Theorem 3. It is not true that for every constructive generalized function \(\Phi_1\) of finite order there exists a number \(n\) such that \(\Phi_1\) is a constructive generalized function of \(n\)-th order.

Theorem 4. There is no algorithm which constructs, for every constructive generalized function \(\Phi_1\) of finite order, a number \(n\) such that \(\Phi_1\) is a constructive generalized function of order \(\leqslant n\).

Theorem 5. There exists an algorithm which transforms every pair of the form \(n \square \Phi_1\), satisfying the condition \(\forall k(|\Phi_1|_k \leqslant n)\), into a word \(\mathfrak m_1\) of type \(\mathfrak m\) such that
\[ \Phi_1=\mathfrak D(n \square R(\mathfrak m_1)). \]
Roughly speaking, this means that every constructive generalized function of order \(\leqslant n\) is the derivative of \(n\)-th order of some locally uniformly continuous function.

Corollary. For every regular constructive generalized function one can construct a word \(\mathfrak m_1\) of type \(\mathfrak m\) such that
\[ \Phi_1=R(\mathfrak m_1). \]

Theorem 6. Whatever the natural number \(n\), there is no algorithm which transforms every constructive generalized function \(\Phi_1\) of order \(\leqslant n\) into a constructive generalized function \(\Phi_2\) such that
\[ \Phi_1=\Phi_2\&\forall k(|\Phi_2|_k\leqslant n). \]

Corollary. There is no algorithm which transforms every regular constructive generalized function \(\Phi_1\) into a word \(\mathfrak m_1\) of type \(\mathfrak m\) such that
\[ \Phi_1=R(\mathfrak m_1). \]

Theorem 3 is proved with the aid of Lemma 2, § 2 from (4). Theorem 6 is proved with the aid of Theorems 10.2.2 and 10.3.1 from (5).

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

Moscow State University
named after M. V. Lomonosov

Received
15 X 1966

CITED LITERATURE

1. N. A. Shanin, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 52, 266 (1958).
2. N. A. Shanin, ibid., 67, 15 (1962).
3. G. S. Tseitin, ibid., 67, 295 (1962).
4. G. S. Tseitin, ibid., 67, 362 (1962).
5. G. E. Mints, ibid., 72, 383 (1964).
6. Fan Dingzhu, DAN, 162, No. 4, 766 (1965).
7. J. Mikusinski, R. Sikorski, The Elementary Theory of Distributions, Warszawa, 1 (1957); 2 (1959).
8. J. Korevaar, Indagation Math., 17, 3, 368; 17, 4, 483; 17, 5, 663 (1955).