MATHEMATICS

K. K. GOLOVKIN

ON CONDITIONS FOR THE SMOOTHNESS OF FUNCTIONS

(Presented by Academician V. I. Smirnov, 1 VI 1960)

1. We shall call any bounded measurable function \(K_n(x)\) satisfying the conditions

\[ \int_{-\infty}^{\infty} K_n(x)x^k\,dx=n!\,\delta_n^k,\qquad \int_{-\infty}^{\infty}|K_n(x)|\,|x|^k\,dx<\infty,\qquad 0\le k\le n. \tag{1} \]

a kernel of order \(n\).

Denote by \(K_n^{(h)}u(x)\) the integral operator

\[ K_n^{(h)}u(x)=\frac1h\int_{-\infty}^{\infty}K_n\left(\frac{y-x}{h}\right)u(y)\,dy. \tag{2} \]

Lemma 1. If \(u(x)\) has continuous bounded derivatives up to order \(n\), then for every kernel \(K_n(x)\) of order \(n\) the equality

\[ \lim_{h\to0}\frac1{h^n}K_n^{(h)}u(x)=u^{(n)}(x) \tag{3} \]

holds.

We shall study classes of summable functions \(u(x)\) satisfying the condition

\[ \sup_x |K_n^{(h)}u(x)|=O(h^\lambda),\qquad \lambda\le n, \tag{4} \]

or, if \(\lambda<n\), the analogous condition in which \(O\) is replaced by \(o\). These classes become normed spaces, which we shall denote respectively by \(K_n^{(\lambda)}\) or \(k_n^{(\lambda)}\), if the norm is defined as
\[ \sup_{x,h}\frac1{h^\lambda}|K_n^{(h)}u(x)|. \]
Along with this, results will be formulated for spaces of functions \(G_{n,p}^{(\lambda)}\) \((\lambda\le n)\) and \(g_{n,p}^{(\lambda)}\) \((\lambda<n)\), having finite norm
\[ \sup_h\frac1{h^\lambda}\|K_n^{(h)}u(x)\|_{L_p}, \]
where for functions from \(g_{n,p}^{(\lambda)}\) the additional condition must hold:
\[ \lim_{h\to0}\frac1{h^\lambda}\|K_n^{(h)}u(x)\|_{L_p}=0. \]

It is proved elementarily:

Lemma 2. The spaces \(k_n^{(\lambda)}\) and \(g_{n,p}^{(\lambda)}\) coincide with the closure, in the corresponding norms, of the set of infinitely differentiable functions.

The norm in \(K_n^{(\lambda)}\) \(\bigl(k_n^{(\lambda)}\bigr)\) is a homogeneous functional having dimension equal to \(-\lambda\) and differential order \(\lambda\) \((^1)\). Therefore it is natural to compare these spaces with the spaces

$C^{(\lambda)}$ ($c^{(\lambda)}$) of sufficiently smooth functions, the norm in which is defined as:

\[ \sup_{x,h}\left|\frac{u^{([\lambda])}(x+h)-u^{([\lambda])}(x)}{h^{\lambda-[\lambda]}}\right| \qquad \text{for fractional } \lambda; \]

\[ \sup_{x,h}\left|\frac{u^{(\lambda-1)}(x+h)-2u^{\lambda-1}(x)+u^{(\lambda-1)}(x-h)}{h}\right| \qquad \text{for integral } \lambda, \]

and the space $c^{(\lambda)}$ coincides with the closure, in these norms, of the infinitely differentiable functions.

Let us clarify some properties of kernels of order $n$. Let $K_n(x)$ be such a kernel. Then the differences $\frac{1}{2^{n+1}}K_n(x/2)-K_n(x)$ and $K_n(x+1)-K_n(x)$ will, after the corresponding normalization, be kernels of order higher than $n$. The following proposition is valid, having the character of the converse to what has been stated.

For every kernel $K_{n+1}(x)$ of order $n+1$ there exist absolutely integrable kernels $A_1(x)$ and $A_2(x)$ and a kernel $K_n(x)$ of order $n$ such that

\[ \frac{1}{2^{n+1}}K_n\left(\frac{x}{2}\right)-K_n(x) = \int_{-\infty}^{\infty} A_1(x-y)K_{n+1}(y)\,dy, \tag{5} \]

\[ K_n(x+1)-K_n(x) = \int_{-\infty}^{\infty} A_2(x-y)K_{n+1}(y)\,dy, \tag{6} \]

\[ \int_{-\infty}^{\infty}|A_1(y)|\,dy+\int_{-\infty}^{\infty}|A_2(y)|\,dy=A. \tag{7} \]

The problem of finding $A_1(x)$, $A_2(x)$, and $K_n(x)$ is easily solved after passing to Fourier transforms.

Lemma 3. If $u(x)\in K_n^{(\lambda)}$ $(k_n^{(\lambda)})$ for $n>\lambda$, then for every integer $m\in(\lambda,n]$ there exists a kernel of order $m$ such that $u(x)$ belongs to the corresponding $K_m^{(\lambda)}$ $(k_m^{(\lambda)})$.

Obviously, it suffices to prove the lemma for $m=n-1$ in the case when this quantity is greater than $\lambda$. Take a kernel $K_m(x)$ satisfying identity (5), and convolve both sides of (5) with $u(xh)$. Dividing the result by $h^\lambda$, we easily arrive at the inequality

\[ \|u\|_{K_m^{(\lambda)}}\le \frac{1}{2^{m-\lambda}}\|u\|_{K_m^{(\lambda)}} + A\|u\|_{K_n^{(\lambda)}} \le \frac{A\|u\|_{K_n^{(\lambda)}}}{1-(1/2)^{m-\lambda}}, \]

which proves the lemma.

Theorem 1. In order that the summable function $u(x)$ belong to $C^{(\lambda)}$ $(c^{(\lambda)})$, it is necessary and sufficient that it belong to some space $K_l^{(\lambda)}$ $(k_l^{(\lambda)})$ with $l>\lambda$.

The necessity of the condition of the theorem is almost trivial. Let us outline the proof of sufficiency, assuming $\lambda$ fractional and beginning with the spaces $c^{(\lambda)}$ and $k_l^{(\lambda)}$. By Lemma 3, we may assume $l=[\lambda]+1$. By Lemma 2, it will be sufficient to establish an estimate of the norm in $c^{(\lambda)}$ in terms of the norm in $k_l^{(\lambda)}$ for $[\lambda]$-times differentiable functions. From the given kernel $K_{[\lambda]+1}(x)$ we construct a kernel $K_{[\lambda]}(x)$ satisfying (5) and (6). Next we write formula (3), putting $n=[\lambda]$ in it, and on the basis of (5) estimate the order of convergence in it, which turns out to be equal to $o(h^{\lambda-[\lambda]})$. After this the difference $u^{[\lambda]}(x+h)-u^{[\lambda]}(x)$ is replaced approximately by the difference $K_{[\lambda]}^{(h)}u(x+h)-K_{[\lambda]}^{(h)}u(x)$, which is estimated on the basis of (6) and also turns out to be of order $o(h^{\lambda-[\lambda]})$. This completes the proof of the required estimate. Now in

in the case of the spaces \(C^{(\lambda)}\) and \(K_l^{(\lambda)}\) we may regard as already proved the existence for \(u(x)\) of a continuous derivative of order \([\lambda]\), since the embedding \(K_l^{(\lambda)} \supset k_l^{(\lambda')}\) holds for every \(\lambda' < \lambda\). Then all the rest of the proof goes through without change. The case of integral \(\lambda\) requires a certain modification of the proof, which is still based only on Lemmas 1—3 and formulas (5) and (6).

Theorem 2. For a summable function \(u(x)\), the conditions \(u(x)\in K_l^{(l)}\) and \(u^{(l-1)}(x)\in \operatorname{Lip} 1\) are equivalent.

Theorem 3. For a summable function \(u(x)\), for any \(n>r\) the conditions \(u(x)\in G_{n,p}^{(r)}\bigl(g_{n,p}^{(r)}\bigr)\), \(u(x)\in H_p^{(r)}\bigl(h_p^{(r)}\bigr)\) are equivalent.

Theorem 4. For a summable function \(u(x)\), the conditions \(u(x)\in G_{l,p}^{(l)}\), \(u(x)\in W_p^{(l)}\) are equivalent.

In proving the last theorem we rely on the equivalence, established by A. A. Dezin \((^2)\), of the conditions
\[ \|u(x+h)-u(x)\|_{L_p}=O(h) \]
and \(u(x)\in W_p^{(1)}\).

Corollary. Let \(\omega(x)\) be a kernel of zero order (an averaging kernel), whose moments from the first to the \((n-1)\)-st are equal to zero, while the \(n\)-th moment is different from zero. Construct the mean of the summable function with respect to this kernel, and let \(h\) be the averaging parameter. Then the smallness of order \(O(h^\lambda)\), \(o(h^\lambda)\) of the norm of the difference \(u(x)-u_h(x)\) in \(L_p\) or \(C\) is equivalent to precisely that smoothness of \(u(x)\) which is guaranteed by Theorems 1—4 under the condition that \(u(x)\) belongs respectively to \(G_{n,p}^{(\lambda)}\) or \(K_n^{(\lambda)}\).

1. Let us write an elementary identity having the form of the Fourier transform of (5), when the role of kernels is played by certain linear combinations of \(\delta\)-functions:
   \[ \frac{1}{2^n}(e^{2i\xi}-1)^n-(e^{i\xi}-1)^n =(e^{i\xi}-1)^{n+1}\sum_{k=0}^{n-1}\frac{1}{2^{k+1}}(e^{i\xi}+1)^k . \tag{8} \]

Identity (8) has as its consequence the coincidence of the coefficients of all powers of \(e^{i\xi}\) in the right- and left-hand sides. We may therefore substitute in (8), in place of the quantities \(e^{i\xi m}\), the values of a completely arbitrary function \(u(x)\) at the points \(x+hm\). This gives us
\[ \Delta_h^{(n)}u(x)=\frac{1}{2^n}\Delta_{2h}^{(n)}u(x) -\sum_{l=0}^{n-1}\alpha_{l,n}\Delta_h^{(n+1)}u(x+hl), \tag{9} \]
where
\[ \alpha_{l,n}=\sum_{k=l}^{n-1}\frac{1}{2^{k+1}}C_k^{(l)},\qquad \Delta_h^{(m)}u(x)=\sum_{k=0}^{m}(-1)^{m-k}C_m^{(k)}u(x+hk). \]

From (9) it obviously follows that

Lemma 4. If \(\|u(x)\|<\infty\), then the conditions \(\|\Delta_h^{(m)}u(x)\|=O(h^\lambda)\) \((o(h^\lambda))\) for all \(m>\lambda\) are equivalent, where \(\|\ \|\) is any functional norm satisfying the condition \(\|u(x+h)\|=\|u(x)\|\) for all \(h\).

From the lemma just proved there follows, in particular, Zigmund’s theorem \((^3)\) on the equivalence of the conditions
\[ |\Delta_h^{(2)}u(x)|=O(h^\lambda) \]
and
\[ |\Delta_h^{(1)}u(x)|=O(h^\lambda) \]
for \(\lambda<1\), which was proved in \((^3)\) by means of the constructive theory of functions. V. A. Solonnikov found an elementary proof of this fact, which influenced our work. With the aid of identity (9) and Lemma 4 one can prove the theorems stated below.

Theorem 5. For a bounded function \(u(x)\), for any \(n>\lambda\) the conditions
\[ \sup_x |\Delta_h^{(n)}u(x)|=O(h^\lambda)\quad (o(h^\lambda)),\qquad u(x)\in C^{(\lambda)}\quad (c^{(\lambda)}) \]
are equivalent.

Theorem 6. For a bounded function \(u(x)\), the conditions
\[ \sup_x \left|\Delta_h^{(n)}u(x)\right|=O(h^n), \qquad u^{(n-1)}(x)\in \operatorname{Lip}1 \]
are equivalent.

Theorem 7. For a function \(u(x)\in L_p\), for any \(n>r\), the conditions
\[ \left\|\Delta_h^{(n)}u(x)\right\|_{L_p}=O(h^r)\ (o(h^r)), \qquad u(x)\in H_p^{(r)}\ (h_p^{(r)}) \]
are equivalent.

Theorem 8. For a function \(u(x)\in L_p\), the conditions
\[ \left\|\Delta_h^{(l)}u(x)\right\|_{L_p}=O(h^l), \qquad u(x)\in W_p^{(l)} \]
are equivalent.

We note that in Theorems 5 and 6 the measurability of \(u(x)\) is not assumed. Theorem 5 follows from a more general result of S. N. Bernstein \({}^{(4)}\).

References Cited

\({}^{1}\) K. K. Golovkin, DAN, 134, No. 1 (1960).
\({}^{2}\) A. A. Dezin, DAN, 88, No. 5 (1953).
\({}^{3}\) A. Zygmund, Duke Math. J., 12 (1945).
\({}^{4}\) S. N. Bernstein, DAN, 57, No. 2 (1947).