Mathematics

Corresponding Member of the Academy of Sciences of the USSR A. N. Tikhonov and A. A. Samarskii

ON THE REPRESENTATION OF LINEAR FUNCTIONALS IN THE CLASS OF DISCONTINUOUS FUNCTIONS

As is known, every linear functional (A[f]), defined in the class (C_0) of continuous functions given on the interval ((a,b)), is represented by means of the Stieltjes integral

[
A[f]=\int_a^b f(x)\,d\alpha(x),
]

where (\alpha(x)) is a function of bounded variation (Riesz theorem). It is also known that this functional can be extended to the class (Q_0) of piecewise-continuous functions. However, this extension is not unique.

The purpose of the present paper is to give a representation for an arbitrary linear functional defined in (Q_0).

§ 1. Consider a linear functional (A[f]), defined by the conditions

1) (A[f_1+f_2]=A[f_1]+A[f_2]);

2) (|A[f]|\leq M\sup |f|)

in the class (Q_0(f)) of piecewise-continuous functions given on the interval ((a,b)).

Consider the piecewise-continuous functions

[
\eta_\xi(x)=
\begin{cases}
1 & \text{if } a<x<\xi;\
0 & \text{if } \xi\leq x<b;
\end{cases}
]

[
\pi_\xi(x)=
\begin{cases}
1 & \text{if } x=\xi;\
0 & \text{if } x\ne \xi;
\end{cases}
]

and denote

[
\alpha(\xi)=A[\eta_\xi(x)], \qquad \sigma(\xi)=A[\pi_\xi(x)].
]

In particular, (\alpha(b)=A[1]), since (\eta_b(x)=1).

In what follows it will be shown that the functions (\alpha(\xi)) and (\sigma(\xi)), which we shall call characteristic, uniquely determine the linear functional on (Q_0).

§ 2. Lemma 1. There exists no more than a countable number of points (\zeta_1,\zeta_2,\ldots,\ldots,\zeta_j,\ldots), at which (\sigma(\xi)\ne 0), and moreover

[
\sum_{j=1}^{\infty} |\sigma(\zeta_j)|\leq M.
]

We shall call a linear functional (A[f]=f(\zeta)\sigma(\zeta)), where (a<\zeta<b), the simplest point functional of the point (\zeta).

A linear functional representable in the form of a sum of simplest point-

... functionals:

[
A[f]=\sum_{j=1}^{\infty}\sigma_j f(\zeta_j)
\left(\sum_{j=1}^{\infty}|\sigma_j|\leq M\right),
]

will be called a point functional.

We shall call the linear functional (\bar A[f]) regular if

[
\bar\sigma(\xi)=\bar A[\pi_\xi(x)]\equiv 0
\quad\text{for every } a<\xi<b.
]

§ 3. We shall show that every linear functional (A[f]) can be represented as the sum of a regular and a point linear functional.

Consider the point functional (A^{*}[f]), equal to

[
A^{*}[f]=\sum_{j=1}^{\infty}\sigma(\zeta_j)f(\zeta_j),
]

where

[
\sigma(\zeta_j)=A[\pi_{\zeta_j}(x)], \qquad
\sum_{j=1}^{\infty}|\sigma(\zeta_j)|\leq M.
]

The linear functional

[
\bar A[f]=A[f]-A^{*}[f],
]

as is not difficult to see, is regular and has norm (\bar M\leq 2M). Hence the possibility of the representation follows:

[
A[f]=\bar A[f]+A^{*}[f].
]

The characteristic function for (A^{*}[f]) is equal to

[
\alpha^{}(\xi)=A^{}[\eta_\xi(x)]
=\sum_{\zeta_j<\xi}\sigma(\zeta_j),
]

and the characteristic function for (\bar A[f]) is given by the formula

[
\bar\alpha(\xi)=\bar A[\eta_\xi(x)]=\alpha(\xi)-\alpha^{*}(\xi).
]

§ 4. Lemma 2. The function (\bar\alpha(\xi)) has bounded variation.

Every function of bounded variation has, at each point, right and left limiting values:

[
\bar\alpha_p(\xi)=\bar\alpha(\xi+0), \qquad
\bar\alpha_l(\xi)=\bar\alpha(\xi-0).
]

Lemma 3. There exists at most a countable number of points
(\xi_1,\xi_2,\ldots,\xi_i,\ldots),
at which
(\bar\alpha_l(\xi_i)\ne\bar\alpha(\xi_i))
or
(\bar\alpha(\xi_i)\ne\bar\alpha_p(\xi_i)).

It should be noted that (\alpha(\xi)) is also a function of bounded variation.

§ 5. Consider the functional

[
\overline{\bar A}[f]
=\bar A[f]
-\sum_{\xi_i<\xi} f_p(\xi_i)\,[\bar\alpha_p(\xi_i)-\bar\alpha(\xi_i)]
-\sum_{\xi_i\leq \xi} f_l(\xi_i)\,[\bar\alpha(\xi_i)-\bar\alpha_l(\xi_i)]
]

and its characteristic function

[
\overline{\bar\alpha}(\xi)
=\overline{\bar A}[\eta_\xi(x)]
=\bar\alpha_p(\xi)
-\sum_{\xi_i<\xi}[\bar\alpha_p(\xi_i)-\bar\alpha_l(\xi_i)].
\tag{1}
]

This formula holds both in the case when (\xi) is a point of discontinuity of the function (\bar{\alpha}(\xi)), and in the case when (\xi) is a point of continuity and (\bar{\alpha}_{\mathrm{п}}(\xi)=\bar{\alpha}(\xi)).

Lemma 4. The function (\bar{\bar{\alpha}}(\xi)) is continuous for (a<\xi<b), and
[
\bar{\bar{\sigma}}(\xi)=\bar{\bar{A}}[\pi_\xi(x)]=0.
]

Let us note that (\bar{\bar{\alpha}}(\xi)) has bounded variation (\bar{\bar{M}}\leq 2\bar{M}).

Lemma 5. If, for some regular functional (\bar{\bar{A}}[f]), the characteristic function (\bar{\bar{\alpha}}(\xi)) is continuous, then in the class (Q_0(f)) the representation
[
\bar{\bar{A}}[f]=\int_a^b f(x)\,d\bar{\bar{\alpha}}(x)
]
holds.

Indeed, divide the interval ((a,b)) into parts by the points (x=x_i), including all points of discontinuity of the function (f(x)), and take the step function
[
\bar{f}(x)=f_{\mathrm{п}}(x_{i-1}),\qquad x_{i-1}<x\leq x_i.
]

The difference (f(x)-\bar{f}(x)) can be represented in the form
[
f(x)-\bar{f}(x)=\varepsilon(x)+\sum_i [f(x_{i-1})-f_{\mathrm{п}}(x_{i-1})]\pi_{x_{i-1}}^{\xi}(x),
]
where (\varepsilon(x)) is a piecewise-continuous function, and (|\varepsilon(x)|<\varepsilon_0) for a sufficiently fine mesh. Hence it follows that
[
\left|\bar{\bar{A}}[f(x)]-\bar{\bar{A}}[\bar{f}(x)]\right|\leq \bar{\bar{M}}\varepsilon_0
\qquad
(\bar{\bar{\sigma}}(x_i)=0)
]
or
[
\left|\bar{\bar{A}}[f]-\sum_i f_{\mathrm{п}}(x_{i-1})
[\bar{\bar{\alpha}}(x_i)-\bar{\bar{\alpha}}(x_{i-1})]\right|
\leq M\varepsilon_0.
]

Passing to the limit as (\Delta x=x_i-x_{i-1}\to 0), we obtain
[
\bar{\bar{A}}[f]
=\lim_{\Delta x\to 0}\sum_i f_{\mathrm{п}}(x_{i-1})
[\bar{\bar{\alpha}}(x_i)-\bar{\bar{\alpha}}(x_{i-1})]
=\int_a^b f(x)\,d\bar{\bar{\alpha}}(x).
]

From the construction of (\bar{\bar{\alpha}}(x)) it is clear that this function is the continuous part of the function (\bar{\alpha}(x)).

§ 6. Thus, the following holds.

Theorem 1. Every linear functional (A[f]), defined in the class (Q_0(f)) of piecewise-continuous functions (f(x)) given on the interval ((a,b)), can be represented in the form
[
A[f]=\int_a^b f(x)\,d\bar{\bar{\alpha}}(x)+
]
[
+\sum_{i=1}^{\infty}{f_{\mathrm{п}}(\xi_i)[\bar{\alpha}{\mathrm{п}}(\xi_i)-\bar{\alpha}(\xi_i)]
+f}}(\xi_i)[\bar{\alpha}(\xi_i)-\bar{\alpha{\mathrm{л}}(\xi_i)]}
+\sum\sigma(\zeta_j)f(\zeta_j),}^{\infty
\tag{2}
]
where
[
\bar{\bar{\alpha}}(\xi)=\alpha(\xi)-\sum_{\zeta_j<\xi}\sigma(\zeta_j);
]
(\alpha(\xi)) and (\sigma(\xi)) are the characteristic functions of the functional (A[f]); (\bar{\bar{\alpha}}(\xi)) is the continuous part of the function (\bar{\alpha}(\xi)), computed by formula (1).

Let us note that for a continuous function

[
\overline{A}[f]=\int_a^b f(x)\,d\overline{\alpha}(x)+
\sum_{i=1}^{\infty} f(\xi_i)\,[\overline{\alpha}\mathrm{p}(\xi_i)-\overline{\alpha}\mathrm{l}(\xi_i)],
]

i.e., the regular functional (\overline{A}[f]) is completely determined by the characteristic function (\alpha(\xi)) at its points of discontinuity.

§ 7. The linear functional

[
\Gamma[f]=\sum_{j=1}^{\infty}\bigl[\omega_j^{(0)} f(\xi_j)+
\omega_j^{(1)} f_\mathrm{l}(\xi_j)+\omega_j^{(2)} f_\mathrm{p}(\xi_j)\bigr],
\quad \text{where } \omega_j^{(0)}+\omega_j^{(1)}+\omega_j^{(2)}=0,
]

will be called a null-functional.

In the class (C_0(f)) a null-functional is always equal to zero. If (\Gamma=\Gamma_R) is a regular functional, then (\omega_j^{(0)}=0), (\omega_j^{(1)}=-\omega_j^{(2)}), and

[
\Gamma_R[f]=\sum_{j=1}^{\infty}\omega_j^{(2)}[f_\mathrm{p}(\xi_j)-f_\mathrm{l}(\xi_j)].
]

Theorem 2. The difference of two linear functionals that coincide on (C_0(f)) is a null-functional on (Q_0(f)).

§ 8. We shall call a linear functional (A[f]) nonnegative (positive) if (A[f]\geq 0) for (f\geq 0) ((A[f]>0) for (f\geq \varepsilon>0)).

Theorem 3. For nonnegativity (positivity) of a linear functional (A[f]), it is necessary and sufficient that the following conditions be satisfied:

1) the characteristic function (\overline{\alpha}(\xi)) of the regular part (\overline{A}[f]) of the functional (A[f]) is a nondecreasing function;

2) the characteristic coefficients (\sigma(\zeta_j)) are nonnegative: (\sigma(\zeta_j)\geq 0) (1), 2) and 3) (\alpha(b)=A[1]>0).

§ 9. We shall call linear regular functionals (A[f]) and (B[f]), defined on (Q_0(f)), mutually symmetric if the condition

[
B[f(x)]=A[f(-x)]
]

is satisfied for any function (f(x)\in Q_0) given on the interval ((a,b)).

Theorem 4. The conditions

[
\alpha(b)=\beta(b), \qquad b=-a, \qquad \beta(\xi)+\alpha(-\xi)=\alpha(b)=\beta(b)
]

are necessary and sufficient for the mutual symmetry of (A[f]) and (B[f]).

§ 10. For some applications a representation of linear functionals on (Q_m(f)) ((m\geq 0)) is required, where (Q_m) is the class of functions piecewise continuous in ((a,b)) together with their derivatives up to order (m) inclusive. Since the characteristic functions (\alpha(\xi)=A[\eta_\xi(x)]) and (\sigma(\xi)=-A[\pi_\xi(x)]) of the functional (A[f]) are determined by means of functions belonging to the class (Q_m), representation (2) holds for (A[f]) defined on (Q_m(f)).

It is not difficult to verify that a linear functional (A), given on (Q_m), can be uniquely extended also to a broader class of functions, for example to the class of functions (R_{\overline{\alpha}(x)}(f)) satisfying the following conditions: 1) (f(x)) is a bounded function measurable on ((a,b)); 2) (f(x)) has right and left limiting values (f_\mathrm{l}) and (f_\mathrm{p}) at all points of discontinuity of the function (\overline{\alpha}(x)).

Received
20 VI 1958