Abstract
Full Text
O. V. BESOV
EXTENSION OF FUNCTIONS WITH PRESERVATION OF DIFFERENTIAL-DIFFERENCE PROPERTIES IN \(L_p\)
(Presented by Academician S. L. Sobolev, 12 I 1963)
Let \([a,b] \subset (a_1,b_1)\), \(f(x)\in L_p(a,b)\), \(1\le p\le \infty\), and let \(\omega_k(f,h)\) be the modulus of smoothness of the function \(f(x)\) with step \(h\) in the metric \(L_p\). In the present note the possibility is established of extending the function \(f(x)\) to the interval \([a_1,b_1]\) by a function \(\varphi(x)\) for which \(\omega_k(\varphi,h)\le C\omega_k(f,h)\), where \(C\) does not depend on the function \(f\). This result is also valid for functions of many variables, and also in the case when, instead of moduli of smoothness of functions, the moduli of smoothness of their derivatives are taken.
For \(k=1\) this result was obtained by V. K. Dzyadyk \((^1)\). For \(k=2\) only separate particular cases were known. The question was solved for functions satisfying the condition \(\omega_2(f,h)\le Mh^r\), with preservation of this property (for \(p=\infty,\ r=1\) by A. F. Timan and V. K. Dzyadyk \((^2)\); for \(p=1,\ r=1\) by V. K. Dzyadyk \((^1)\); for \(1\le p\le \infty,\ 0<r<2\) by O. V. Besov \((^3)\)), and also for functions satisfying the condition
\[ \int_{0}^{\varepsilon} h^{-1-r\theta}\omega_2^\theta(f,h)\,dh<\infty \]
(for \(1\le p=\theta<\infty\) by V. P. Il’in and V. A. Solonnikov \((^4)\); for \(1\le p\le \infty,\ 1\le \theta<\infty\) independently by O. V. Besov \((^3)\)). In a note by K. K. Golovkin and V. A. Solonnikov \((^7)\) a theorem is given on the extension of functions with preservation of more general conditions connected with the concept of a functional of maximization type. For arbitrary moduli of smoothness of second order with \(p=\infty\), the estimate \(\omega_2(\varphi,h)\le 5\omega_2(f,h)\) was obtained by V. K. Dzyadyk \((^8)\) and T. Freem \((^9)\).
The method of consideration refines the method proposed in \((^3)\); it is close to V. P. Il’in’s method \((^4)\) of integral representation of functions.
Theorem 1. Let \(f(x)\in L_q(0,a)\) have a generalized (in the sense of Sobolev) derivative \(f^{(k)}(x)\),
\[ \omega_m\bigl(f^{(k)},h\bigr)_{L_p(0,a)} = \sup_{0<t\le h} \left\| \sum_{\nu=0}^{m}(-1)^{m-\nu}\binom{m}{\nu} f^{(k)}(x+\nu t) \right\|_{L_p(0,a-mt)}, \]
\[ 1\le q\le \infty,\quad 1\le p\le \infty,\quad 0<a\le \infty. \]
Then there exists a function \(\varphi(x)\in L_q(-a,a)\), coinciding with the function \(f(x)\) on \([0,a]\), and such that
\[ \|\varphi\|_{L_q(-a,a)}\le C\|f\|_{L_q(0,a)}, \tag{1} \]
\[ \omega_m\bigl(\varphi^{(k)},h\bigr)_{L_p(-a,a)} \le C\omega_m\bigl(f^{(k)},h\bigr)_{L_p(0,a)}, \tag{2} \]
where the constant \(C\) does not depend on the function \(f\).
For the proof, consider the function \(F(x,y)\), which is the \(m\)-th Steklov mean for the function \(f(x)\):
\[ F(x,y)=y^{-m}\int_{0}^{y}\cdots\int_{0}^{y} f(x+t_1+\cdots+t_m)\,dt_1\cdots dt_m. \tag{3} \]
We note that
\[ F_x^{(m+k)}(x,y)=y^{-m}\Delta^m[f^{(k)}(x),y], \tag{4} \]
\[ \Delta_x^m[\Phi_x^{(k)}(x,y),h] = \int_0^h\cdots\int_0^h \Phi^{(m+k)}(x+t_1+\cdots+t_m,y)\,dt_1\cdots dt_m, \tag{5} \]
\[ \Delta_y^m[F_x^{(k)}(x,0),y] = \sum_{\nu=0}^{m}(-1)^{m-\nu}\binom{m}{\nu}F_x^{(k)}(x,\nu y) = \]
\[ =(-1)^mF_x^{(k)}(x,0)+ \sum_{\nu=1}^{m}(-1)^{m-\nu}\binom{m}{\nu}(\nu y)^{-m} \int_0^{\nu y}\cdots\int_0^{\nu y} f^{(k)}(x+t_1+\cdots+t_m)\,dt_1\cdots dt_m = \]
\[ = \sum_{\nu=0}^{m}(-1)^{m-\nu}\binom{m}{\nu}y^{-m} \int_0^y\cdots\int_0^y f^{(k)}(x+\nu(\xi_1+\cdots+\xi_m))\,d\xi_1\cdots d\xi_m, \]
whence
\[ \Delta_y^m[F_x^{(k)}(x,0),y] = y^{-m}\int_0^y\cdots\int_0^y \Delta^m[f^{(k)}(x),\xi_1+\cdots+\xi_m]\,d\xi_1\cdots d\xi_m. \tag{6} \]
From the equality
\[ \Delta_y^m\Delta_x^m\Phi_x^{(k)}(x,0) = \Delta_x^m\Delta_y^m\Phi_x^{(k)}(x,0) \]
we obtain that
\[ \Delta_x^m[\Phi_x^{(k)}(x,0),y] = \sum_{\nu=1}^{m}\alpha_\nu\Delta_x^m[\Phi_x^{(k)}(x,\nu y),y] + \sum_{\nu=0}^{m}\beta_\nu\Delta_y^m[\Phi_x^{(k)}(x+\nu y,0),y]. \tag{7} \]
The function \(F(x,y)\), constructed by formula (3), is defined for
\(0\le x\le a-my\). Consider the extension of the function \(F(x,y)\) across the \(y\)-axis by the Whitney and Hestenes method
\[ \Phi(x,y)= \begin{cases} F(x,y), & 0\le x\le a-my,\\[4pt] \displaystyle\sum_{\nu=1}^{m+k}\lambda_\nu F\!\left(-\frac{x}{\nu},y\right), & 0\le -x\le a-my, \end{cases} \]
where
\[ \sum_{\nu=1}^{m+k}\left(-\frac{1}{\nu}\right)^s\lambda_\nu=1 \quad\text{for } s=0,1,\ldots,m+k-1. \]
We shall show that the function \(\varphi(x)=\Phi(x,0)\), \(|x|\le a\), is the desired extension of the function \(f(x)\). By virtue of formulas (7), (5), (6), for \(0<my<2\delta=\dfrac{a}{m+1/2}\) we have
\[ \|\Delta^m[\varphi^{(k)}(x),y]\|_{L_p(-\delta,\delta-my)} = \|\Delta_x^m[\Phi_x^{(k)}(x,0),y]\|_{L_p(-\delta,\delta-my)} \le \]
\[ \le c_1\sum_{\nu=1}^{m} \left\| \int_0^y\cdots\int_0^y \Phi_x^{(m+k)}(x+t_1+\cdots+t_m,\nu y)\,dt_1\cdots dt_m \right\|_{L_p(-\delta,\delta-my)} + \]
\[ + c_2\|\Delta_y^m[F_x^{(k)}(x,0),y]\|_{L_p(0,\delta)} \le c_3\sum_{\nu=1}^{m}y^m \|\Phi_x^{(m+k)}(x,\nu y)\|_{L_p(-\delta,\delta)} + \]
\[ + c_4\left\| y^{-m}\int_0^y\cdots\int_0^y \Delta^m[f^{(k)}(x),\xi_1+\cdots+\xi_m]\,d\xi_1\cdots d\xi_m \right\|_{L_p(0,\delta)} \le \]
\[ \leqslant c_5 \sum_{\nu=1}^{m} y^m \left\| F_x^{(m+k)}(x,\nu y)\right\|_{L_p(0,\delta)} + c_4 y^{-1}\left\|\int_0^{my}\left|\Delta^m\left[f^{(k)}(x),\xi\right]\right|\,d\xi\right\|_{L_p(0,\delta)} \]
\[ \leqslant c_5 \sum_{\nu=1}^{m} \nu^{-m}\left\|\Delta^m\left[f^{(k)}(x),\nu y\right]\right\|_{L_p(0,\delta)} + c_6 y^{-1}\int_0^{my}\left\|\Delta^m\left[f^{(k)}(x),\xi\right]\right\|_{L_p(0,\delta)}\,d\xi \leqslant \]
\[ \leqslant c_7 \omega_m\left(f^{(k)},y\right)_{L_p(0,a)}. \]
This proves the theorem.
Corollary. Let \(\varepsilon>0\), \(1\leqslant p\leqslant \infty\), and let \(f(x)\in L_p(a,b)\) have a generalized derivative \(f^{(k)}(x)\). Then there exists a finite function \(\varphi(x)\in L_p(-\infty,\infty)\), coinciding with the function \(f(x)\) on \([a,b]\), equal to zero outside the interval \([a-\varepsilon,b+\varepsilon]\), and such that
\[ \|\varphi\|_{L_p(-\infty,\infty)} + h^{-m}\omega_m\left(\varphi^{(k)},h\right)_{L_p(-\infty,\infty)} \leqslant \]
\[ \leqslant C\left\{\|f\|_{L_p(a,b)} + h^{-m}\omega_m\left(f^{(k)},h\right)_{L_p(a,b)}\right\}, \tag{8} \]
where \(C\) does not depend on the function \(f\).
Extend the function \(f(x)\) to the interval \([a-\varepsilon,b+\varepsilon]\) by Theorem 1 and denote it by the same symbol. Consider the function \(\varphi(x)=f(x)\eta(x)\), where \(\eta(x)\) is infinitely differentiable and \(\eta(x)=1\) for \(a-\varepsilon/3\leqslant x\leqslant b+\varepsilon/3\), while \(\eta(x)=0\) outside the interval \([a-2\varepsilon/3,b+2\varepsilon/3]\). The estimate of the first term on the left-hand side of formula (8) is obvious.
In what follows, by the norm in \(L_p\) we shall mean the norm in the sense of \(L_p(\alpha,\beta)\), where \([\alpha,\beta]\subset(a-\varepsilon,b+\varepsilon)\). In the course of obtaining inequalities, when necessary, we shall pass to larger intervals from \((a-\varepsilon,b+\varepsilon)\), without mentioning this each time. This is permissible, since it is enough to prove estimate (8) for all \(h\subset(0,h_0]\), where \(h_0\) is arbitrarily small:
\[ \left\|\Delta^m\left[\varphi^{(k)},h\right]\right\|_{L_p} \leqslant c_1\left\|\Delta^m\left[\sum_{s=0}^{k}\eta^{(k-s)}f^{(s)},h\right]\right\|_{L_p} \leqslant \]
\[ \leqslant c_2\sum_{\nu=0}^{m}\sum_{s=0}^{k} h^\nu \left\|\Delta^{m-\nu}\left[f^{(s)},h\right]\right\|_{L_p}. \tag{9} \]
Let us note that
\[ h^{-\nu}\omega_\nu(\psi,h)_{L_p} \leqslant C\left\{\|\psi\|_{L_p}+h^{-m}\omega_m(\psi,h)_{L_p}\right\}, \qquad 1\leqslant \nu\leqslant m . \]
A proof of this fact for \(p=\infty\) is given in \((^5)\), p. 118; for \(1\leqslant p<\infty\) it is carried out analogously.
We shall also use the relation
\[
\|\Delta[\psi,h]\|_{L_p}\leqslant h\|\psi'\|_{L_p}.
\]
From inequality (9) we shall have
\[ h^{-m}\omega_m\left(\varphi^{(k)},h\right)_{L_p(-\infty,\infty)} \leqslant c_3 \sum_{s=0}^{k} \left\{ \|f^{(s)}\|_{L_p} + h^{-m-k+s}\omega_{m+k-s}\left(f^{(s)},h\right)_{L_p} \right\} \leqslant \]
\[ \leqslant c_4\left\{ \sum_{s=0}^{k}\|f^{(s)}\|_{L_p} + h^{-m}\omega_m\left(f^{(k)},h\right)_{L_p} \right\}. \]
To prove inequality (8), it remains to show that
\[ \|f^{(s)}\|_{L_p} \leqslant c\left\{ \|f\|_{L_p} + h^{-m}\omega_m\left(f^{(k)},h\right)_{L_p} \right\}, \qquad s=1,2,\ldots,k. \tag{10} \]
We shall use an idea of S. M. Nikol’skii \((^6)\), p. 269. Write the expansion of the function \(f(x)\) by Taylor’s formula
\[ f(x+\nu h) = \sum_{s=0}^{k-1}\frac{f^{(s)}(x)}{s!}(\nu h)^s + \frac{1}{(k-1)!}\int_0^{\nu h}(\nu h-\xi)^{k-1}f^{(k)}(x+\xi)\,d\xi = \]
\[ = \sum_{s=0}^{k-1}\frac{f^{(s)}(x)}{s!}(\nu h)^s + \frac{\nu^k}{(k-1)!}\int_0^{h}(h-\xi)^{k-1}f^{(k)}(x+\nu\xi)\,d\xi . \]
Hence
\[ \sum_{s=1}^{k}\frac{f^{(s)}(x)}{s!}(\nu h)^s = f(x+\nu h)-f(x) - \frac{\nu^k}{(k-1)!}\int_{0}^{h}(h-\xi)^{k-1}\,[f^{(k)}(x+\nu\xi)-f^{(k)}(x)]\,d\xi . \]
Multiplying by \((-1)^{m-\nu}\binom{m}{\nu}\nu^{-k}\) and summing over \(\nu\), we shall have
\[ \sum_{s=1}^{k} h^s \sum_{\nu=1}^{m}(-1)^{m-\nu}\binom{m}{\nu}\nu^{s-k}\frac{f^{(s)}(x)}{s!} = \]
\[ = \sum_{\nu=1}^{m}[f(x+\nu h)-f(x)](-1)^{m-\nu}\binom{m}{\nu}\nu^{-k} - \]
\[ - \frac{1}{(k-1)!}\int_{0}^{h}(h-\xi)^{k-1}\Delta^{m}[f^{(k)}(x),\xi]\,d\xi . \tag{11} \]
Choosing in (11) successively \(h=h_i,\ i=1,2,\ldots,k\), where the \(h_i\) are sufficiently small positive distinct numbers, we obtain a system of algebraic equations with a Vandermonde determinant, from which one can obtain an expression for \(f^{(k)}(x)\) in terms of the right-hand sides. Hence also
\[ \|f^{(k)}\|_{L_p} \le c_1\left\{\|f\|_{L_p}+\int_{0}^{h_0}\|\Delta^{m}[f^{(k)}(x),\xi]\|_{L_p}\,d\xi\right\} \le \]
\[ \le c_2\left\{\|f\|_{L_p}+h_0^{-m}\omega_m(f^{(k)},h_0)_{L_p}\right\} \le 2^m c_2\left\{\|f\|_{L_p}+h^{-m}\omega_m(f^{(k)},h)_{L_p}\right\}, \]
where \(0<h\le h_0\), \(h_0\) is sufficiently small (see \((^5)\), p. 116). By the same device it is even easier to show that
\[ \|f^{(s)}\|_{L_p}\le c_3\{\|f\|_{L_p}+\|f^{(k)}\|_{L_p}\},\qquad s=1,2,\ldots,k-1. \]
Hence, and from the preceding inequality, we obtain (10), and at the same time the proof of the corollary. Let us also note that in the estimate for \(h^{-m}\omega_m(\varphi^{(k)},h)_{L_p(-\infty,\infty)}\) the first term on the right-hand side of inequality (8) is essential, as is easily checked for \(f(x)=\text{const}\). Analogously one proves
Theorem 2. Let a function \(f(x_1,\ldots,x_n)\) be given on the parallelepiped
\[ \Delta\equiv\{a_i\le x_i\le b_i,\ i=1,\ldots,n\} \]
and have there a generalized (in the sense of S. L. Sobolev) derivative \(\partial^k f/\partial x_1^k\) with partial modulus of smoothness in the direction of the axis \(x_1\)
\[ \omega_m\left(\frac{\partial^k f}{\partial x_1^k},he_1\right)_{L_p(\Delta)} . \]
Then it can be extended by the Whitney–Hestenes method to a larger parallelepiped
\[ \Delta_1\equiv\{a'_1\le x_1\le b'_1,\ a_i\le x_i\le b_i,\ i=2,\ldots,n\} \]
as a function \(\varphi(x_1,\ldots,x_n)\), in such a way that
\[ \omega_m\left(\frac{\partial^k\varphi}{\partial x_1^k},he_1\right)_{L_p(\Delta_1)} \le c\,\omega_m\left(\frac{\partial^k f}{\partial x_1^k},he_1\right)_{L_p(\Delta)} . \]
For Theorem 2 there is also valid a corollary analogous to the corollary to Theorem 1. The corollary to Theorem 1 admits generalization to the case of an \(n\)-dimensional domain whose boundary locally satisfies a Lipschitz condition.
Received
2 I 1963
CITED LITERATURE
- V. K. Dzyadyk, Mat. sborn., 40 (82), 239 (1956).
- A. F. Timan, V. K. Dzyadyk, DAN, 75, No. 4, 499 (1950).
- O. V. Besov, Mat. sborn., 58 (100), No. 1, 87 (1962).
- V. P. Il’in, V. A. Solonnikov, DAN, 136, No. 3, 538 (1961).
- A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.
- S. M. Nikol’skii, Mat. sborn., 33 (75), 2, 261 (1953).
- K. K. Golovkin, V. A. Solonnikov, DAN, 143, No. 4, 767 (1962).
- V. K. Dzyadyk, DAN, 121, No. 3, 403 (1958).
- T. Frey, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., III, 8/1, 89 (1958).