UDC 517.511

MATHEMATICS

Ya. B. LIVCHAK

ON ONE APPROACH TO THE CONCEPT OF LIMIT

(Presented by Academician A. N. Tikhonov, 10 VII 1968)

Our aim is to construct a theory of limits in such a way that every sequence has a limit.

Let \(S\) be the vector space consisting of all real countable sequences. We identify the real number \(a\) with the corresponding stationary sequence, i.e., with the sequence \(x=\{x_n\}\) such that \(x_n=a\) for every \(n\). The set \(\Gamma\) of all sequences converging to zero is a subspace in \(S\). The limit of a sequence \(x\) is equal to the number \(a\) if and only if the cosets \(\Gamma+a\) and \(\Gamma+x\) coincide. This circumstance allows us to regard the coset \(\Gamma+x\) as the limit of the sequence \(x=\{x_n\}\) from \(S\): \(\Gamma+x=\operatorname{Lim} x_n\). We identify the real number \(a\) with the coset \(\Gamma+a\). If the sequence \(x\) converges and \(a=\lim x_n\), then \(a=\Gamma+a=\Gamma+x\), i.e., the introduced definition of limit does not contradict the classical one. The coset \(\Gamma+x\) exists for every \(x\) from \(S\), i.e., every sequence has a limit \(\Gamma \operatorname{Lim} x_n\). On the basis of these preliminary considerations we shall construct a theory of limits. From the example considered it is clear that the limit \(\Gamma \operatorname{Lim} x_n\) of a sequence of real numbers may be not a real number, but an element of some set containing the number line.

Let \(I\) be a right-filtering partially ordered set, and let to each index \(i\in I\) there correspond an (additive) group \(G_i\); denote the complete direct sum of these groups by \(\Sigma^{*}G_i\). If all \(G_i\) coincide, then we denote \(G_i=G\), \(\Sigma^{*}G_i=G^I\). We shall call an \(I\)-indexed sequence an element \(\{a_i\}\) of \(\Sigma^{*}G_i\). Take a normal divisor \(H\) in the group \(\Sigma^{*}G_i\). By the limit \(H\operatorname{Lim} a_i\) modulo \(H\) of an \(I\)-indexed sequence \(a=\{a_i\}\) from \(\Sigma^{*}G_i\) we shall mean the coset \(H+a\). If all \(G_i\) are equal to one another and equal to \(G\), then denote the factor group \((\Sigma^{*}G_i)/H\) by \(G_H^I\). The limit of a sum is equal to the sum of the limits. If all \(G_i\) are vector spaces over a field \(K\), and \(H\) is a subspace in the space \(\Sigma^{*}G_i\), then \(H\operatorname{Lim}\alpha x_i=\alpha H\operatorname{Lim}x_i\). If all \(G_i\) are rings and \(H\) is an ideal in the ring \(\Sigma^{*}G_i\), then
\(H\operatorname{Lim}(x_i y_i)=(H\operatorname{Lim}x_i)(H\operatorname{Lim}y_i)\).

We shall say that a normal divisor \(H\) of \(\Sigma^{*}G_i\) separates a subset \(M\) of \(\Sigma^{*}G_i\) if every coset modulo \(H\) contains no more than one element of \(M\). If \(H\) separates \(M\) and the class \(H+a\) contains an element \(m\) of \(M\), then we identify the coset \(H+a\) with \(m\). In particular, if \(M\) is the number line, \(m\in M\), \(m\in H\operatorname{Lim}a_i=H+a\), then we shall regard the limit \(H\operatorname{Lim}a_i\) as the real number \(m\). An \(I\)-indexed sequence \(\{x_i\}\) from \(\Sigma^{*}G_i\) will be called almost zero if there exists an index \(i_0\in I\) such that \(x_i=0\) as soon as \(i\ge i_0\). The set \(T\) of all almost zero sequences is a normal divisor in the group \(\Sigma^{*}G_i\). The limit \(T\operatorname{Lim}x_i\) will be called the fine limit. The group \((\Sigma^{*}G_i)/T\) is the reduced product of the groups \(G_i\). A sequence \(\{a_i\}\) will be called stationary if there exists \(a\) from \(G=\bigcap G_i\) such that \(a_i=a\) for every \(i\) from \(I\); since \(T\) separates \(G\), we can identify \(a\) with \(T+\{a_i\}\). If all \(G_i\) are rings, then \(\Sigma^{*}G_i\) is also a ring, and \(T\) is a two-sided ideal in the ring \(\Sigma^{*}G_i\), therefore
\(T\operatorname{Lim}(x_i y_i)=(T\operatorname{Lim}x_i)(T\operatorname{Lim}y_i)\). Let all \(G_i\) be vector structures. Put \(\{x_i\}\cap\{y_i\}=\{x_i\cap y_i\}\), \(\{x_i\}\cup\{y_i\}=\{x_i\cup y_i\}\). Then

\(\Sigma^*G_i\) is a vector structure. If, moreover, \(H\) is an \(l\)-ideal in \(\Sigma^*G_i\), then the factor group \((\Sigma^*G_i)/H\) is also a vector structure, and
\(H \operatorname{Lim} (x_i \cup y_i) = (H \operatorname{Lim} x_i) \cup (H \operatorname{Lim} y_i)\),
\(H \operatorname{Lim} (x_i \cap y_i) = (H \operatorname{Lim} x_i) \cap (H \operatorname{Lim} y_i)\),
\(|H \operatorname{Lim} x_i| = H \operatorname{Lim} |x_i|\), and from \(x \geqslant y\) it follows that
\(H \operatorname{Lim} x_i \geqslant H \operatorname{Lim} y_i\).
Suppose that each vector structure \(G_i\) contains the number line \(R\) as a vector substructure. A sequence \(\{t_i\}\) from \(\Sigma^*G_i\) will be called infinitesimal if, for any positive \(\varepsilon\) from \(R\), there exists an index \(i_\varepsilon\) from \(I\) such that \(|x_i| < \varepsilon\) as soon as \(i \geqslant i_\varepsilon\). The set \(\Gamma\) of all infinitesimal sequences is an \(l\)-ideal in the vector structure \(\Sigma^*G_i\). The limit \(\Gamma \operatorname{Lim} x_i\) will be called the rough limit. \(\Gamma\) separates \(R\).

Denote by \(N\) the set of all natural numbers, and by \(R^m\) the real \(m\)-dimensional vector space with fixed basis \(e_1,\ldots,e_m\). For any \(n\) from \(N\), define the set \(A_n^m\) (a half-open cube), assigning a vector \(x=(x_1,\ldots,x_m)\) from \(R^m\) to \(A_n^m\) if \(-n \leqslant x_i < n\) \((1 \leqslant i \leqslant m)\). For any \(n\) from \(N\) and any \(x\) from \(R\), there exists a unique number \(r_n(x)\) from \(R\) such that \(-n \leqslant r_n(x) < n\) and \(x \equiv r_n(x) \bmod 2n\), i.e. the difference \(x-r_n(x)\) is an integer multiple of \(2n\). We have defined the function \(r_n(x)\) on \(R\) with values in \(A_n^1\). Construct the mapping \(r_n^m(x)\) of the space \(R^m\) onto the half-open cube \(A_n^m\). For any \(x=(x_1,\ldots,x_m)\), put \(r_n^m(x)=(r_n(x_1),\ldots,r_n(x_m))\). Introduce on the cube \(A_n^m\) the operation \(\oplus\), setting, for any \(x,y\) from \(A_n^m\), \(x \oplus y = r_n^m(x+y)\). \(A_n^m\) is a group. On \(A_n^m\) the natural topology is introduced, and \(A_n^m\) is a bicompact topological group (the \(m\)-dimensional torus). Lebesgue measure on the cube \(A_n^m\) is Haar measure. Real measurable functions on the cube \(A_n^m\) are regarded, as usual, as equal if they coincide almost everywhere. Let \(F_n^m\) be the set of all real bounded measurable functions on \(A_n^m\). The thin limit
\[ f(x)=T \operatorname{Lim}_n f_n(x) \]
from \((\Sigma^*F_n^m)/T\) will be called a generalized function on \(R^m\).

A real function \(\varphi(x_1,\ldots,x_k)\) on \(R^k\) will be called locally bounded if it is bounded on every bounded subset of \(R^k\). Let \(\varphi(x_1,\ldots,x_k)\) be a Borel locally bounded function on \(R^k\), and let generalized functions
\[ f_1(x)=T \operatorname{Lim}_n f_n^1(x),\ldots, f_k(x)=T \operatorname{Lim}_n f_n^k(x) \]
be given on \(R^m\). Define the generalized function \(\varphi(f_1(x),\ldots,\)
\[ \ldots,f_k(x)), \]
putting
\[ \varphi(f_1(x),\ldots,f_k(x)) = T \operatorname{Lim}_n \varphi(f_n^1(x),\ldots,f_n^k(x)). \]
Since the function \(\varphi(x,y)=xy\) on \(R^2\) is Borel and locally bounded, the product of any pair of generalized functions on \(R^m\) is defined.

Let a series
\[ \sum_{i=1}^{\infty} a_i x^i \]
and a generalized function \(f(x)=T \operatorname{Lim}_n f_n(x)\) on \(R^m\) be given. The sum
\[ \sum_{i=1}^{\infty} a_i f^i(x) = T \operatorname{Lim}_n \sum_{i=1}^{n} a_i f_n^i(x) \]
is also a generalized function on \(R^m\). Let \(\psi\) be a measurable mapping of the space \(R^m\) into itself, i.e. for any measurable subset \(E\) of \(R^m\) its preimage is measurable. For any \(n\) from \(N\), define the mapping \(\psi_n\) of the cube \(A_n^m\) into itself by putting
\[ \psi_n(x)=r_n^m(\psi(x)). \]
If \(f(x)=T \operatorname{Lim}_n f_n(x)\) is a generalized function on \(R^m\), then, for any \(n\) from \(N\), \(f_n(\psi_n(x)) \in F_n^m\). The thin limit \(T \operatorname{Lim}_n f_n(\psi_n(x))\) is a generalized function. Denote
\[ f(\psi(x))=T \operatorname{Lim}_n f_n(\psi_n(x)). \]
Let \(f(x)=T \operatorname{Lim}_n f_n(x)\) be a generalized function on \(R^m\), \(i\) an integer, \(1 \leqslant i \leqslant m\). For any \(n\) from \(N\), the measurable function on \(A_n^m\)
\[ \partial_n f_n(x)/\partial x_i = n\bigl(f_n(x \oplus n^{-1}e_i)-f_n(x)\bigr) \]
is bounded, i.e. the limit \(T \operatorname{Lim}_n(\partial_n f_n(x)/\partial x_i)\) is a generalized function on \(R^m\). Denote
\[ \partial f/\partial x_i = T \operatorname{Lim}_n(\partial_n f_n(x)/\partial x_i). \]
For \(m=1\) we denote
\[ \partial f(x)/\partial x = df(x)/dx = f'(x). \]

For any \(n\) from \(N\), define the cube \(V_n^m\) as the set of all vectors from \(R^m\) such that \(0<x_1<n^{-1},\ldots,0<x_m<n^{-1}\). Denote by \(\Phi_n^m\) the set of all real bounded measurable functions on the cube \(V_n^m\), \(\Phi^m=(\sum_n^* \Phi_n^m)/T\), \(A^m=(\sum_n^* A_n^m)/T\). Let \(f(x)=T\operatorname{Lim}_n f_n(x)\) be a generalized function on \(R^m\). We associate with it a function \(f^*(x)\) on \(A^m\). Take a point \(\alpha=T\operatorname{Lim}_n \alpha_n\) from \(A^m\) and construct the value \(f^*(\alpha)\). For any \(n\) from \(N\), construct a function \(f_n(t_n,\alpha_n)\) on \(V_n^m\), which we shall regard as a function of \(t_n\), setting, for any \(t_n\) from \(V_n^m\),
\(f_n(t_n,\alpha_n)=f_n(t_n\oplus \alpha_n)\). The limit \(T\operatorname{Lim}_n f_n(t_n,\alpha_n)\) from \(\Phi^m\) will be regarded as the value of the function \(f^*(x)\) at the point \(\alpha\) and denoted
\(f^*(T\operatorname{Lim}_n \alpha_n)=T\operatorname{Lim}_n f_n(t_n,\alpha_n)\). If \(f(x)=T\operatorname{Lim}_n f_n(x)\), \(g(x)=T\operatorname{Lim}_n g_n(x)\) are generalized functions on \(R^m\) and, for any \(\alpha=T\operatorname{Lim}_n \alpha_n\) from \(A^m\), \(f^*(\alpha)=g^*(\alpha)\), then the generalized functions \(f(x)\) and \(g(x)\) coincide. Therefore we may identify a generalized function \(f(x)\) on \(R^m\) with the function \(f^*(x)\) on \(A^m\). In this sense every generalized function \(f(x)\) on \(R^m\) is an ordinary function on \(A^m\) with values in \(\Phi^m\). To each generalized function \(f(x)=T\operatorname{Lim}_n f_n(x)\) on \(R^m\) there correspond the integrals

\[ T\int f\,d\mu = T\operatorname{Lim}_n \int_{A_n^m} f_n(x)\,d\mu \quad\text{and}\quad \Gamma\int f\,d\mu = \Gamma\operatorname{Lim}_n \int_{A_n^m} f_n(x)\,d\mu . \]

Let \(\varphi(x)\) be a real measurable locally bounded function on \(R^m\). Denote

\[ T\int f(x)\varphi(x)\,d\mu = T\operatorname{Lim}_n \int_{A_n^m} f_n(x)\varphi(x)\,d\mu, \qquad \Gamma\int f(x)\varphi(x)\,d\mu \]

\[ = \Gamma\operatorname{Lim}_n \int_{A_n^m} f_n(x)\varphi(x)\,d\mu . \]

Let \(f(x)=T\operatorname{Lim}_n f_n(x)\) and \(g(x)=T\operatorname{Lim}_n g_n(x)\) be generalized functions on \(R^m\).

Denote

\[ h(x)=f(x)*g(x)=T\operatorname{Lim}_n \int_{A_n^m} f_n(\xi)g_n(x-\xi)\,d\xi, \]

and call the generalized function thus obtained the convolution. All the preceding operations are also applicable to complex generalized functions
\(f(x)=T\operatorname{Lim}_n(\alpha_n(x)+i\beta_n(x))\).
Let \(f(x)=T\operatorname{Lim}_n f_n(x)\) be a complex generalized function on \(R^m\). For any \(n\) from \(N\) and any \(\lambda=(\lambda_1,\ldots,\lambda_m)\) from \(A_n^m\), denote

\[ c_n(\lambda)=\int_{A_n^m} e^{-2\pi i\lambda x} f_n(x)\,dx, \]

where \(\lambda x=\lambda_1x_1+\cdots+\lambda_mx_m\).

The generalized function \(c(\lambda)=T\operatorname{Lim}_n c_n(\lambda)\) will be called the Fourier transform of the generalized function
\(f(x)=T\operatorname{Lim}_n f_n(x)\).

Let \(\varphi(x_1,\ldots,x_{k+2})\) be a locally bounded Borel function on \(R^{k+2}\), and let generalized functions
\(f_1(x)=T\operatorname{Lim}_n f_n^1(x),\ldots,f_k(x)=T\operatorname{Lim}_n f_n^k(x)\) be given on \(R\). We shall say that the generalized function
\(y(x)=T\operatorname{Lim}_n y_n(x)\) is a solution of the differential equation

\[ dy(x)/dx=\varphi(x,y(x),f_1(x),\ldots,f_k(x)) \tag{1} \]

on the interval \(a\le x\le b\), if for every \(x\) from this interval relation (1) is satisfied. Choose a point
\(y_0=T\operatorname{Lim}_n y_n^0(t)\) from \(\Phi^m\), and we shall seek a generalized function \(y(x)=T\operatorname{Lim}_n y_n(x)\) on \(R\) satisfying relation (1) for \(a\le x\le b\) and the initial condition \(y(a)=y_0\). Choose \(n_0\) from \(N\) so that \(n_0>|a|+1\) and \(n_0>|b|+1\). For \(n<n_0\) put \(y_n(x)=0\). Let \(n\ge n_0\). For \(-n<x<a\) put \(y_n(x)=0\). For \(a<x<a+n^{-1}\) put
\(y_n(x)=y_n^0(x-a)\). For \(a+n^{-1}<x<a+2n^{-1}\) put

\[ y_n(x)=y_n(x-n^{-1})+ n\varphi\bigl(x-n^{-1},y(x-n^{-1}),f_1(x-n^{-1}),\ldots,f_k(x-n^{-1})\bigr). \tag{2} \]

For \(a+2n^{-1}<x<a+3n^{-1}\) we define \(y_n(x)\) by formula (2), and so on. There exists \(k_n\in N\) such that
\[ a+n^{-1}k_n\leq b<a+n^{-1}(k_n+1). \]
For \(a+n^{-1}(k_n+1)<x<n\) we put \(y_n(x)=0\). The generalized function
\[ y(x)=\Gamma \operatorname{Lim} y_n(x) \]
is the unique solution of the problem posed.

Suppose generalized functions \(f(x)\) and \(g(x)\) are given on \(R\). The equation
\[ y'(x)=f(x)g(x)y(x) \tag{3} \]
is a special case of equation (1). We note that similar equations (with a four-dimensional argument \(x\)) are used in quantum electrodynamics \({}^{1}\), and that, in the known presentations of the theory of generalized functions, solving equation (3) is connected with difficulties.

Let \(f(x,y)\) be a Borel function on \(R^2\), let \(y(x)\) be an ordinary solution of the equation \(y'=f(x,y)\) on the interval \(a\leq x\leq b\), suppose that the Lipschitz condition is satisfied for all \(y\) and \(z\), and that along the curve \(y=y(x)\), for \(a\leq x\leq b\), the derivative \(df(x,y(x))/dx\) is bounded. If
\[ z(x)=\Gamma \operatorname{Lim} z_n(x) \]
is a generalized solution of the equation \(y'=f(x,y)\), then the sequence \(\{z_n(x)\}\) converges uniformly on the interval \(a\leq x\leq b\) to the function \(y(x)\).

Put \(e(x)=0\) for \(-n<x<0\), \(e(x)=1\) for \(0<x<n\); \(\delta(x)=e'(x)\). If \(\varphi(x)\) is a measurable function on \(R\) and, for large \(n\), the integrals
\[ \int_{-n^{-1}}^{0}\varphi(x)\,dx \quad\text{and}\quad \int_{n-n^{-1}}^{n}\varphi(x)\,dx \]
exist, then
\[ \Gamma\int \delta(x)\varphi(x)\,dx = \Gamma\operatorname{Lim}_{n}\, n\int_{-n^{-1}}^{0}\varphi(x)\,dx - \Gamma\operatorname{Lim}_{n}\, n\int_{n-n^{-1}}^{n}\varphi(x)\,dx, \]
\[ \Gamma\int \delta^{2}(x)\varphi(x)\,dx = \Gamma\operatorname{Lim}_{n}\, n^{2}\int_{-n^{-1}}^{0}\varphi(x)\,dx + \Gamma\operatorname{Lim}_{n}\, n^{2}\int_{n-n^{-1}}^{n}\varphi(x)\,dx. \]

If the finite function \(\varphi(x)\) has a finite first derivative at zero, then
\[ \Gamma\int \delta^{2}(x)\varphi(x)\,dx = \varphi(0)\Gamma\operatorname{Lim}_{n} n-\frac{1}{2}\varphi'(0). \]

Latvian State University
named after P. Stučka

Received
6 VII 1967

References Cited

\({}^{1}\) P. K. Rashevskii, UMN, 13, no. 3 (81), 3 (1958).