MATHEMATICS

S. M. NIKOL’SKII

A REPRESENTATION THEOREM FOR A CLASS OF DIFFERENTIABLE FUNCTIONS OF SEVERAL VARIABLES BY MEANS OF ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

(Presented by Academician I. M. Vinogradov on 25 XII 1962)

In this note we consider a class of functions which I shall denote by
\(S_p^{(r)}H(R_n)^*\).

Let \(n\) be a natural number; \(e_n=\{1,\ldots,n\}\) is the set of natural numbers (indices) not exceeding \(n\); \(e\subset e_n\) is an arbitrary subset of it. In particular, \(e\) may be the empty set \((e=0)\) or may coincide with \(e_n\). Further, let \(\mathbf r=(r_1,\ldots,r_n)\) be a vector (with nonnegative coordinates). Denote by \(e_r\) the support of \(\mathbf r\), i.e. the set of indices \(j\) for which \(r_j>0\). Thus, \(r_j>0\) for \(j\in e_r\) and \(r_j=0\) for \(j\in e_n-e_r\).

Denote by \(\mathbf r^e=(r_1^e,\ldots,r_n^e)\) a vector such that \(r_j^e=r_j\) for \(j\in e\) and \(r_j^e=0\) for \(j\in e_n-e\). Let also \(\omega_n=(1,\ldots,1)\), and let \(R_n\) be the \(n\)-dimensional real space of points \(\bar x=(x_1,\ldots,x_n)\).

By definition, a function \(f(\bar x)\) belongs to the class
\(S_p^{(r)}H(R_n)=S_p^{(r)}\) \((1\le p\le \infty)\) if the following properties hold:

1) \(f\in L_p(R_n)\).

2) The generalized derivatives \(f^{(\mathbf r^e)}\) make sense on \(R_n\), where \(\mathbf r=(r_1,\ldots,r_n)\) and \(r_j=\bar r_j+\alpha_j\), \(\bar r_j\) is an integer, \(0<\alpha_j\le 1\).

3)
\[ \frac{\left\|\Delta_{\mathbf h}^{2\omega_n^e}f^{(\mathbf r^e)}\right\|_{L_p(R_n)}}{\prod_{j\in e}|h_j|^{\alpha_j}} \le M_p^{(\mathbf r^e)}(f)<\infty \tag{1} \]
for every \(e\subset e_r\) and vector \(\mathbf h=(h_1,\ldots,h_n)\) with \(h_j>0\) \((j=1,\ldots,n)\).

\(\Delta_{\mathbf h}^{2\omega_n^e}\) denotes the multiple difference of the second order in \(x_j\) with steps \(h_j\) for \(j\in e\). The constant \(M_p^{(\mathbf r^e)}(f)\) does not depend on \(\mathbf h\).

We put
\[ \|f\|_{S_p^{(r)}}=\|f\|_{S_p^{(r)}H(R_n)} =\sum_{e\subset e_r} M_p^{(\mathbf r^e)}(f), \qquad M_p^{(\mathbf r^0)}(f)=\|f\|_{L_p(R_n)}. \]

Theorem 1. A function \(f\in S_p^{(r)}H(R_n)=S_p^{(r)}\) \((1\le p\le \infty)\) can be represented in the form
\[ f(\bar x)=\sum_{e\subset e_r}\sum_{\mathbf k^e\ge 0} Q_{\mathbf k^e}(\bar x), \tag{2} \]

* This class is, in a certain sense, close to the class \(S_p^{(r)}\), which was considered in my note \((^8)\).

where the first sum extends over all subsets \(e \subset e_r\), including \(e=0\), \(e=e_r\). The second sum extends over all possible integral nonnegative vectors \(\mathbf{k}^e=(k_1^e,\ldots,k_n^e)\) \((k_j^e=0\) for \(j\in e_n-e)\). The functions \(Q_{\mathbf{k}^e}\) are entire functions of exponential type of degrees \(2^{k_j^e}\) in \(x_j\), \(j\in e_r\) (thus, of degree 1 in \(x_j\), \(j\in e_r-e\)), satisfying the inequalities

\[ \|Q_{\mathbf{k}^e}\|_{L_p(R_n)} \le \frac{c\|f\|_{S_p^{(r)}H(R_n)}}{2^{\sum_{j\in e} k_j^e r_j}}, \qquad (e\subset e_r). \tag{3} \]

Theorem 2 (converse to Theorem 1). Let a function \(f(\bar{x})\), defined on \(R_n\), be representable in the form (2) with terms \(Q_{\mathbf{k}^e}\) possessing the properties listed in Theorem 1, where in (3) it is only necessary to replace \(\|f\|_{S_p^{(r)}H(R_n)}\) by \(M\).

Then \(f\in S_p^{(r)}H(R_n)\) and the inequality

\[ \|f\|_{S_p^{(r)}(R_n)}\le cM, \]

holds, where \(c\) does not depend on \(M\).

The proof of Theorem 1 is based on the introduction of an entire function \(g(t)\) of degree 1, for which

\[ \int_{-\infty}^{\infty} g(t)\,dt=1,\qquad \int_{-\infty}^{\infty} |t|^s |g(t)|\,dt\le b^\rho \quad (s=0,1,\ldots,\rho), \]

where \(\rho>0\) is a given natural number. Let \(m\) be a natural number, \(1\le m\le n\), and let \(\vec{\omega}_m=(\underbrace{1,\ldots,1}_{m\ \text{times}})\). Put, for \(\vec{\mu}=(\mu_1,\ldots,\mu_m)\) \((\mu_j>0)\),

\[ \sigma_{\vec{\mu}}(\bar{x}) = \int_{-\infty}^{\infty}\cdots \int_{-\infty}^{\infty} \prod_{j=1}^{m} g(u_j)\, \Delta^{(\rho+2)\vec{\omega}_m}_{\frac{u_1}{\mu_1},\ldots,\frac{u_m}{\mu_m}} f(\bar{x})\,du_1\cdots du_m, \tag{4} \]

where the multiple difference has order \(\rho+2\) in each variable \(x_1,\ldots,x_m\), respectively, with steps

\[ \frac{u_1}{\mu_1},\ldots,\frac{u_m}{\mu_m}. \]

Next, if we set

\[ \tau_{\mathbf{k}}(\bar{x}) = \Delta^{\vec{\omega}_m}_{1,\ldots,1}\, \sigma_{2^{k_1},\ldots,2^{k_m}} \qquad (k_j=0,1,\ldots;\ j=1,\ldots,m), \]

where \(\Delta^m_{1,\ldots,1}\) denotes the \(m\)-fold difference of first order in each of the variables \(k_j\) with step 1, then it turns out that

\[ f(\bar{x}) = \sum{}' \gamma_{\mathbf{i}}(x) + \sum_{\mathbf{k}\ge 0}\tau_{\mathbf{k}}(\bar{x}), \qquad 0\le \mathbf{i}\le(\rho+2)\vec{\omega}_m,\quad \mathbf{k}=(k_1,\ldots,k_m), \tag{5} \]

where \(\sum{}'\) extends over all integral nonnegative, but nonzero, vectors \(\mathbf{i}\) for which \(0\le \mathbf{i}\le(\rho+2)\vec{\omega}_m\), and the sum \(\sum_{\mathbf{k}}\) is an infinite \(m\)-fold series.

If a function \(f\in L_p(R_n)\) is such that

\[ \|\sigma_{\vec{\mu}}\|_p\to 0 \quad\text{as}\quad \sum_{1}^{m}\mu_j^2\to 0, \tag{6} \]

then the series \(\sum_{\mathbf{k}}\) converges in the sense of \(L_p(R_n)\). For finite \(p\), condition (6) follows automatically from the fact that \(f\in L_p(R_n)\).

The following properties hold:

1) \(\|\gamma_i\|_p,\quad \|\tau_{\mathbf{k}}\|_p \le c\|f\|_p.\)

2) \(\gamma_i(\bar{x})\) is entire of degree 1 in \(x_j\), \(j\in e_i\), and in those \(x_j\), \(j\in e_n-e^m\), with respect to which \(f\) is entire of degree 1.

3) \(\tau_k\) are entire of degrees \(2^{k_j}\) in \(x_j\), \(j\in e_m\), and of degree 1 in those \(x_j\), \(j\in e_n-e_m\), with respect to which \(f\) is entire of degree 1.

4) If \(f\in S_p^{(r)}\), \(e_r\subset e_m\), \(0\leq r\leq \rho\vec{\omega}_m\), then \(\gamma_i\in S_p^{r(e_r-e_i)}\) and
\[ \|\gamma_i\|_{S_p^{r(e_r-e_i)}}\leq c\|f\|_{S_p^{(r)}} . \]

5) If \(f\in S_p^{(r)}\), \(e_r\subset e_m\), \(0<r\leq \rho\vec{\omega}_m\), then
\[ \|\tau_k\|_p\leq \frac{c\|f\|_{S_p^{(r)}}}{2^{\sum_{j\in e_r} k_j r_j}} . \tag{7} \]

To obtain Theorem 1, one must apply equality (5) with \(e_r=e_m\), and then from \(f(\bar{x})\) there is separated off the sum \(\sum_k\), which in (2) corresponds to \(e\in e_r\). To the functions \(\gamma_i\) Theorem 1 is again applied, where the role of \(e_m\) must now be played by the set \(e_r-e_i\). As a result, for each \(i\) a series \(\sum_{k^e}\tau_{k^e}\) with \(e=e_r-e_i\) is separated off. After a finite number of operations the process of separation terminates and leads to the series (2).

Theorem 2 is proved in the spirit of the converse theorem of approximation of S. N. Bernstein. We give several inequalities that can be obtained as consequences of Theorems 1 and 2.

If \(1\leq p\leq p'\leq \infty\), \(\vec{\rho}=(\rho_1,\ldots,\rho_n)=r-\left(\frac1p-\frac1{p'}\right)\vec{\omega}_n\), \(\rho_j>0\) \((j=1,\ldots,n)\), then*
\[ S_p^{(r)}(R_n)\to S_p^{(\vec{\rho})}(R_n). \tag{8} \]

If
\[ e^1+e^2=e_r,\qquad e^1e^2=0,\qquad e_n=e_r+e^3,\qquad r_j-\frac1p>0,\quad j\in e^2, \]
then
\[ S_p^{(r)}(R_n)\to S_p^{(r e^1+e^3)}(R_{e^1+e^3}), \tag{9} \]
where \(R_{e^1+e^3}\) is the subspace of points \(\bar{x}=(x_1,\ldots,x_n)\) with fixed \(x_j=x_j^0\), \(j\in e^2\).

Explanation. Represent the function \(f\in S_p^{(r)}(R_n)\) in the form (2), and estimate the norms \(\|Q_{k^e}\|_{p'}\), by already known inequalities (see \(\left({}^{1}\right)\) (3) or \(\left({}^{2}\right)\) (1,10)), through the norms \(\|Q_{k^e}\|_p\), whose estimates are expressed by inequalities (3). Then we apply the converse theorem and obtain (8). We proceed in the same way in the case (9), but now the \(\|Q_{k^e}\|_{L_p(R_{e^1+e^3})}\) are estimated by other known inequalities through \(\|Q_{k^e}\|_{L_p(R_n)}\) (see \(\left({}^{2}\right)\), (1.12)). It is also necessary to replace all the series (2) by series of smaller multiplicity, summing them with respect to the indices \(k_j^e\), \(j\in e^2\).

The following assertion holds:

If \(f\in S_p^{(r)}(R_n)S_p^{(\vec{\rho})}(R_n)\), \(\lambda+\mu=1\), \(\lambda,\mu\geq 0\), then \(f\in S_p^{(\lambda r+\mu\vec{\rho})}(R_n)\) and
\[ \|f\|_{S_p^{(\lambda r+\mu\vec{\rho})}} \leq c\|f\|_{S_p^{(r)}}^\lambda \|f\|_{S_p^{(\vec{\rho})}}^\mu . \tag{10} \]

* For normed spaces \(E,E_1\) we write \(E\to E_1\) if \(E\subset E_1\), and \(\|f\|_{E_1}\leq c\|f\|_E\) for all \(f\in E\).

Explanation. The theorem (10) is first proved when \(e_r=e_{\vec\rho}=e'\), where \(e'\) consists of a single index. Suppose now that (10) holds when \(e_r,e_{\vec\rho}\subset e_{m-1}\), and it is given that \(e_r+e_{\vec\rho}\subset e_m=e_{\lambda r+\mu\vec\rho}\). We apply to \(f\) the expansion (5), in which, therefore, simultaneously

\[ \|\tau_k\|_\rho \leq \frac{c_1\|f\|_{S_\rho^{(r)}}}{2^{\sum_{j\in e_r} k_j r_j}}, \qquad \|\tau_k\|_\rho \leq \frac{c_1\|f\|_{S_\rho^{(\vec\rho)}}}{2^{\sum_{j\in e_{\vec\rho}} k_j \rho_j}}. \tag{11} \]

Raising the first and second inequalities respectively to the powers \(\lambda,\mu\), and then multiplying, we obtain

\[ \|\tau_k\|_\rho \leq \frac{c_1\|f\|_{S_\rho^{(r)}}^\lambda \|f\|_{S_\rho^{(\vec\rho)}}^\mu} {2^{\sum_{j\in e_m} k_j(\lambda r_j+\mu\rho_j)}}. \]

As a result, the sum of the series with terms \(\tau_k\), by Theorem 2, belongs to
\(S_\rho^{(\lambda r+\mu\vec\rho)}\). Further (by property 4)),

\[ \gamma_i \to \in S_\rho^{(r e_r-e_i)} S_\rho^{(\vec\rho e_{\vec\rho}-e_i)}, \]

\((e_r-e)+(e_{\vec\rho}-e_i)\subset e_m-e_i\), and since \(e_i\neq0\), because \(i\neq0\), it follows, in view of the fact that the assertion is true for \(m-1\), that

\[ f\in S_\rho^{(q)}, \tag{12} \]

where

\[ q=\lambda r^{(e_r-e_i)}+\mu\rho^{(e_{\vec\rho}-e_i)} =(\lambda r+\mu\rho)^{(e_m-e_i)}. \]

Moreover, \(f\) is an entire function of degree 1 with respect to \(x_j,\ j\in e_i\), and this, together with (12), implies that \(f\in S_\rho^{\lambda r+\mu\vec\rho}\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
17 XII 1962

REFERENCES

1. S. M. Nikol’skii, DAN, 76, 785 (1951).
2. S. M. Nikol’skii, Tr. Mat. Inst. im. V. A. Steklov, Academy of Sciences of the USSR, 38, 244 (1951).
3. S. M. Nikol’skii, DAN, 146, No. 3 (1962).
4. S. M. Nikol’skii, UMN, 16, 5 (101), 63 (1961).