E. V. MAIKOV

\(\tau\)-POLYNOMIALS, \(\tau\)-ANALYTIC FUNCTIONALS

(Presented by Academician P. S. Aleksandrov on 23 I 1964)

In \((^1)\) the concept of \(\tau\)-differentiability was introduced and the general form was found of a functional whose \((n+1)\)-st \(\tau\)-derivative is identically zero. Such functionals were called \(\tau\)-polynomials. Here we shall consider some of their properties; in particular, we shall obtain for them an expansion analogous to the Taylor polynomial. Then, using this expansion, we shall give the definition of \(\tau\)-analyticity and formulate some properties of \(\tau\)-analytic functionals.

The space \(\Xi\). As in \((^1)\), \(\xi(\tau)\) is a piecewise-continuous function on \(D=[0;1]\) with values in \(R=(-\infty,+\infty)\) (a trajectory), such that \(\xi(\tau+0)=\xi(\tau)\), \(\xi(1-0)=\xi(1)\). \(\Xi=\{\xi(\tau)\}\). Elements of \(\Xi\) are denoted by \(\xi\), possibly with an upper index. \(\xi_\beta^\alpha\) will always denote \(\xi^\alpha(t_\beta)\). \(f(\xi)\) is a complex-valued functional on \(\Xi\).

\(\tau\)-Polynomial. In \((^1)\) it was established that the set of \(\tau\)-polynomials coincides with the set of functionals representable in the form

\[ f(\xi)=\int_0^1 \cdots \int_0^1 \varphi(t_1,\xi_1,\ldots,t_n,\xi_n)\,dt_1\ldots dt_n =\int \varphi\,dt^n, \tag{1} \]

where \(\varphi(t_1,x_1,\ldots,t_n,x_n)\) is an arbitrary continuous complex-valued function of \(2n\) real variables (the density of the functional), defined on \((D\times R)^n\); \(\xi_j=\xi(t_j)\) (in accordance with the notation introduced above); \(t^n\) is Lebesgue measure on \(D^n\). In what follows we shall everywhere assume that the density \(\varphi\) is a symmetric function of pairs of its arguments, i.e., for any substitution \(p_n=(\alpha_1,\ldots,\alpha_n)\) of the numbers \((1,\ldots,n)\),

\[ \varphi(t_{\alpha_1},x_{\alpha_1},\ldots,t_{\alpha_n},x_{\alpha_n}) = \varphi(t_1,x_1,\ldots,t_n,x_n). \]

This does not restrict generality: if \(\varphi\) in (1) is not symmetric, then, denoting

\[ S_n\varphi=\frac{1}{n!}\sum_{p_n} \varphi(t_{\alpha_1},x_{\alpha_1},\ldots,t_{\alpha_n},x_{\alpha_n}), \]

we obtain \(f(\xi)=\int S_n\varphi\,dt^n\), where \(f(\xi)\) is the same as in (1), while \(S_n\varphi\) is symmetric.

The totality of all \(\tau\)-polynomials is a linear ring, which we shall denote by \(P\).

The representation of \(f(\xi)\in P\) in the form (1) is not unique. For example, the functional (1) will not change if to its density \(\varphi\) one adds a continuous symmetric function \(\theta(t_1,\ldots,t_n)\) (not depending on \(x_j\)) such that its mean value on \(D^n\) is equal to zero. Likewise, if one takes \(\widetilde{\varphi}=S_{n+1}\varphi\), then the functional (1) is equal to \(f(\xi)=\int \widetilde{\varphi}\,dt^{n+1}\).

Canonical expansion of a functional \(f(\xi)\in P\). Consider an arbitrary function \(\varphi(t_1,x_1,\ldots,t_n,x_n)\), \(t_j\in D\), \(x_j\in R\). Fix \(\xi^0\in\Xi\). We shall call \(\varphi\) simple relative to the trajectory \(\xi^0\) (more briefly, \(\xi^0\)-simple) if for any \(j=1,\ldots,n\)

\[ \varphi(t_1,x_1,\ldots,t_{j-1},x_{j-1},t_j,\xi_j^0,t_{j+1},x_{j+1},\ldots,t_n,x_n)\equiv 0. \]

We shall call \(\varphi\) degenerate if it can be represented in the form \(\varphi=\sum_{j=1}^n \varphi_j\), where the \(j\)-th summand does not depend on \(x_j\).

Remark 1. If the density \(\varphi\) of the functional \(f(\xi)\) in (1) is degenerate, then it can be represented in the form of an integral over \(D^{n-1}\): \(f(\xi)=\int \tilde{\varphi}\,dt^{\,n-1}\), where \(\tilde{\varphi}\) is a continuous function on \((D\times R)^{n-1}\).

For brevity, introduce the notation

\[ \varphi [x_{11}\pm x_{12}\pm\cdots]\cdots [x_{n1}\pm x_{n2}\pm\cdots] \tag{2} \]

(\(\pm\) denotes either plus or minus), which we shall understand as follows: first, regarding (2) as a product, expand the square brackets, preserving the order of the factors, and then, instead of each of the resulting terms of the form \(\pm\varphi x_{1\beta_1}\cdots x_{n\beta_n}\), write \(\pm\varphi(t_1,x_{1\beta_1},\ldots,t_n,x_{n\beta_n})\). For example:
\[ \varphi[x_1]\cdots[x_n]=\varphi(t_1,x_1,\ldots,t_n,x_n), \]
\[ \varphi[x_1-\xi_1][x_2]\cdots[x_n] =\varphi(t_1,x_1,t_2,x_2,\ldots,t_n,x_n) -\varphi(t_1,\xi_1,t_2,x_2,\ldots,t_n,x_n). \]

Denote
\[ H\varphi=\varphi[x_1-\xi_1^0]\cdots[x_n-\xi_n^0]. \]

Lemma 1. For \(\varphi\) to be \(\xi^0\)-simple, it is necessary and sufficient that \(\varphi=H\varphi\).

Lemma 2. If \(H\varphi\equiv 0\) for some \(\xi^0\), then \(\varphi\) is degenerate. From the degeneracy of \(\varphi\) it further follows that \(H\varphi\equiv0\) for any \(\xi^0\in\Xi\).

Corollary. If \(\varphi\) is simultaneously simple and degenerate, then \(\varphi\equiv0\).

Theorem 1. Every function \(\varphi(t_1,x_1,\ldots,t_n,x_n)\) can be represented uniquely in the form \(\varphi=\varphi^1+\varphi^2\), where \(\varphi^1\) is \(\xi^0\)-simple and \(\varphi^2\) is degenerate. Moreover,
\[ \varphi^1=H\varphi,\qquad \varphi^2=\varphi-H\varphi. \]

Example 1. Let \(n=2\), \(\varphi=(x_1+x_2)^3\), \(\xi^0\equiv0\). Then
\[ \varphi^1=3x_1x_2(x_1+x_2),\qquad \varphi^2=x_1^3+x_2^3. \]

Example 2. If \(n=2\), \(\varphi=\cos(x_1+x_2)\), \(\xi^0\equiv0\), then
\[ \varphi^1=\cos(x_1+x_2)-\cos x_1-\cos x_2+1,\qquad \varphi^2=\cos x_1+\cos x_2-1. \]

From Theorem 1 and Remark 1 it follows:

Theorem 2. Any \(\tau\)-polynomial (1) can be represented in the form
\[ f(\xi)=\int \psi^n(t_1,\xi_1,\ldots,t_n,\xi_n)\,dt^n+\cdots+\int \psi^1(t_1,\xi_1)\,dt^1+\psi^0, \tag{3} \]
where \(\psi^0=f(\xi^0)\), and \(\psi^1(t_1,x_1),\ldots,\psi^n(t_1,x_1,\ldots,t_n,x_n)=H\varphi\) are \(\xi^0\)-simple functions; moreover, for \(j=1,\ldots,n-1\),
\[ \psi^j=\binom{j}{n}\int_{D^{\,n-j}}\varphi[x_1-\xi_1^0]\cdots[x_j-\xi_j^0][\xi_{j+1}^0]\cdots[\xi_n^0]\,dt_{j+1}\cdots dt_n. \tag{4} \]

Representation (3) will be called the canonical expansion of the \(\tau\)-polynomial \(f(\xi)\) about the trajectory \(\xi^0\); the largest number \(s\) with non-identically-zero function \(\psi^s\) (\(s\le n\)) is the degree of the \(\tau\)-polynomial, and the functions \(\psi^0,\ldots,\psi^s\) are its canonical coefficients.

Simple \(\tau\)-polynomial. If a \(\tau\)-polynomial can be represented in the form
\[ f(\xi)=\int \psi^s\,dt^s, \]
where \(\psi^s\) is a \(\xi^0\)-simple function, then it is called simple relative to \(\xi^0\). Each summand on the right-hand side of (3) is a \(\xi^0\)-simple \(\tau\)-polynomial. (The latter is true if \(\xi^0(t)\) is continuous. In the case of discontinuous \(\xi^0\), it may turn out that the function \(\psi^j\) in (3) does not satisfy the continuity condition imposed by us on the density of a \(\tau\)-polynomial. However, this condition itself is not so essential.)

Theorem 3. If \(f(\xi)\) is a \(\xi^0\)-simple \(\tau\)-polynomial, then its \(\tau\)-derivatives (see (1)) satisfy the equalities
\[ f^{(j)}(\xi^0,t_1,x_1,\ldots,t_j,x_j)\equiv0 \qquad \text{for } j=0,1,\ldots,s-1, \]
\[ f^{(s)}(\xi^0,t_1,x_1,\ldots,t_s,x_s)=s!\,\psi^s(t_1,x_1,\ldots,t_s,x_s). \]

(Thus, a simple \(\tau\)-polynomial is analogous to the function \((z-z^0)^s\), where \(z\) is a complex variable.)

Theorem 4. The expansion (3) of an arbitrary $\tau$-polynomial into a sum of $\xi^0$-simple $\tau$-polynomials is unique (despite the nonuniqueness of the notation (1)).

Indeed, as follows from Theorem 3, the $j$-th canonical coefficient of a $\tau$-polynomial is determined by the value of its $j$-th $\tau$-derivative at $\xi=\xi^0$:

\[ \psi^j=\frac{1}{j!} f^{(j)}(\xi^0,t_1,x_1,\ldots,t_j,x_j). \tag{5} \]

If $\psi_1^1,\ldots,\psi_1^m$ and $\psi_2^1,\ldots,\psi_2^n$ $(m\le n)$ are the canonical coefficients, respectively, of the $\tau$-polynomials $f_1$ and $f_2$, then the canonical coefficients of the sum $f_1+f_2$ have the form

\[ \psi^j=\psi_1^j+\psi_2^j \quad (j=0,1,\ldots,n,\ \psi_1^j\equiv 0 \text{ for } j>m), \]

and the canonical coefficients of the product $f_1\cdot f_2$ have the form

\[ \psi^j=S_j\sum_{k=0}^{j}\psi_1^k(t_1,x_1,\ldots,t_k,x_k)\cdot \psi_2^{j-k}(t_{k+1},x_{k+1},\ldots,t_j,x_j), \]

where $j=0,1,\ldots,m+n$.

Taylor’s formula for a $\tau$-polynomial. Using equalities (5), the canonical expansion (3) of a $\tau$-polynomial can be rewritten in the form

\[ f(\xi)=f(\xi^0)+\sum_{j=1}^{s}\frac{1}{j!} \int f^{(j)}(\xi^0,t_1,\xi_1,\ldots,t_j,\xi_j)\,dt^j . \tag{6} \]

We shall call equality (6) the Taylor expansion of the $\tau$-polynomial $f(\xi)$ about the trajectory $\xi^0$.

Let us note that equalities (4) and (5) give an algebraic method for computing the $\tau$-derivative of any order of a $\tau$-polynomial, if its density $\varphi$ is known.

Example 3. Let

\[ f(\xi)=\int \cos(\xi_1+\xi_2)\,dt^2,\qquad \xi^0\equiv 0. \]

Then $f(0)=1$,

\[ f^{(1)}(0,t_1,x_1)=2\int_{0}^{1}(\cos x_1-1)\,dt_2 =2(\cos x_1-1), \]

\[ f^2(0,t_1,x_1,t_2,x_2)= \cos(x_1+x_2)-\cos x_1-\cos x_2+1. \]

$\tau$-Analytic functionals. Let $f(\xi)$ be an arbitrary complex-valued functional on $\Xi$, infinitely $\tau$-differentiable for every $\xi^0\in\Xi$. If

\[ f(\xi)=f(\xi^0)+\sum_{j=1}^{\infty}\frac{1}{j!} \int f^{(j)}(\xi^0,t_1,\xi_1,\ldots,t_j,\xi_j)\,dt^j, \tag{7} \]

then $f(\xi)$ is called $\tau$-analytic on $\Xi$, and the right-hand side of (7) is its Taylor series.

Example 4. Every $\tau$-polynomial is $\tau$-analytic.

Example 5. The functional

\[ f(\xi)=\exp\left(\int_{0}^{1} v(\xi(t))\,dt\right), \]

where $v(x)$ is a continuous function on $R$, is $\tau$-analytic (but does not belong to $P$). Its $j$-th $\tau$-derivative is equal to

\[ f^{(j)}= \exp\left[\int_{0}^{1} v(\xi^0(t))\,dt\right]\cdot \prod_{k=1}^{j}\bigl(v(x_k)-v(\xi_k^0)\bigr). \tag{8} \]

If we suppose that \(v(0)\equiv 0\) and take \(\xi^0=0\), then instead of (8) we obtain

\[ f^j(0,t_1,x_1,\ldots,t_j,x_j)=v(x_1)\cdots v(x_n). \]

The corresponding Taylor series has the form

\[ f(\xi)=1+\sum_{j=1}^{\infty}\frac{1}{j!}\int v(\xi_1)\cdots v(\xi_j)\,dt^j . \]

The ring \(M\) of \(\tau\)-analytic functionals. Take a sequence
\(\varphi^0,\varphi^1(t_1,x_1),\ldots,\varphi^j(t_1,x_1,\ldots,t_j,x_j),\ldots\), where \(\varphi^j\) is a continuous symmetric function on \((D\times R)^j\), simple with respect to \(\xi^0=0\), and satisfying the inequality \(|\varphi^j|\le a^{j+1}\), where \(0<a<\infty\). Consider the series

\[ \varphi^0+\sum_{j=1}^{\infty}\frac{1}{j!}\int \varphi^j(t_1,\xi_1,\ldots,t_j,\xi_j)\,dt^j=f(\xi), \]

which converges for all \(\xi\) and defines a \(\tau\)-analytic functional \(f(\xi)\), with
\(f^{(j)}(0,t_1,x_1,\ldots,t_j,x_j)=\varphi^j\).

Denote by \(M(a)\) the totality of all such functionals, and put
\(M=\bigcup_a M(a)\). (The functional from Example 5 belongs to \(M\), if \(v(x)\) is bounded on \(R\).)

Introduce in \(M\) a countable sequence of norms

\[ \|f\|_n=\max_{0\le j\le n}\ \sup_{\xi,t_k,x_k} \left|f^{(j)}(\xi,t_1,x_1,\ldots,t_j,x_j)\right|, \]

where \(n=0,1,2,\ldots\).

Take a sequence \(f_1,\ldots,f_m,\ldots\in M\). We shall say that it converges to \(f\in M\) in the space \(M\), if there exists such an \(a\) that all \(f_m\in M(a)\), and for every \(n\)

\[ \lim_{m\to\infty}\|f-f_m\|_n=0. \]

Then the limit of the sequence also belongs to \(M(a)\).

Some properties of the space \(M\).

1. \(M\) is a linear ring.
2. Every \(f(\xi)\in M\) is bounded: if \(f(\xi)\in M(a)\), then \(|f(\xi)|\le ae^a\).
3. If \(f(\xi)\in M(a)\subset M\), then \(f(\xi+\xi^0)\in M(2ae^a)\subset M\) for any continuous \(\xi^0\in \Xi\).
4. The ring \(M\) is complete with respect to convergence in \(M\).
5. Every \(f(\xi)\in M\) is equal, in the sense of convergence in \(M\), to the sum of its Taylor series.
6. \(P_M=P\cap M\) is everywhere dense in \(M\).
7. If \(\{f_m\}\) converges to \(f\) in the space \(M\), then for every \(\xi\) the equality
   \[ \lim_{m\to\infty} f_m(\xi)=f(\xi) \]
   holds.

The convergence introduced in \(M\) turns this space into a union of countably normed spaces. It was used by the author for a generalization of the concept of measure. Elements of the space conjugate to \(M\) we call \(\tau\)-measures. The concept of a \(\tau\)-measure turns out to be convenient in considering such systems of finite-dimensional distributions in a functional space which cannot be extended to an ordinary measure in it (see (2)).

I express my gratitude to Prof. S. V. Fomin for useful discussions on topics connected with this work.

Moscow State University
named after M. V. Lomonosov

Received
15 X 1963

CITED LITERATURE

1. E. V. Maikov, DAN, 155, No. 2 (1964).
2. E. V. Maikov, UMN, 18, issue 3, 243 (1963).