Reports of the Academy of Sciences of the USSR
1970. Volume 195, No. 5

MATHEMATICS

Yu. N. SUBBOTIN

APPROXIMATION OF FUNCTIONS OF THE CLASS \(W^kH_\omega^p\) BY SPLINES OF ORDER \(m\)

(Presented by Academician I. M. Vinogradov on 18 V 1970)

Let \(f(x)\in W_p^k\), i.e., it has an absolutely continuous \((k-1)\)-st derivative and

\[ \|f^{(k)}(x)\|_{L_p(0,1)} = \left\{\int_0^1 |f^{(k)}(x)|^p\,dx\right\}^{1/p}<\infty \qquad (1\le p\le \infty). \tag{1} \]

In this paper we consider questions of approximation of such functions on \([0,1]\) by splines \(S_{m,n}(x)\) of order \(m\) \((m\ge k)\) with a number of knots \(\{x_s\}\) not exceeding \(n\), in the metric \(L_q\). The case \(m=k-1\) was considered in \((^1)\). The spline \(S_{m,n}(x)\) is a piecewise-polynomial function, glued from polynomials of order not higher than \(m\) in such a way that the derivative \(S_{m,n}^{(m-1)}(x)\) is continuous on \([0,1]\), while \(S_{m,n}^{(m)}(x)\) has discontinuities only at the knots \(\{x_s\}\). The cases of fixed and nonfixed placement of knots are considered. For \(p=q=\infty\), for fixed knots \(\{x_s\}\) and odd splines, there is a rather extensive literature devoted to these questions (for a bibliography see, for example, \((^2)\)); there, under a certain relation between \(k\) and \(m\), the case \(p=2,\ q=\infty\) is also considered. We also note an interesting result of V. M. Tikhomirov \((^3)\), pertaining to the case \(p=q=\infty\). In the present paper we also consider approximations of functions \(f(x)\), defined on the whole real axis and belonging to the class \(W^kH_\omega^p\), by interpolating splines \(S_m(x,h)\) \((^{4,5})\) of order \(m\), which interpolate the function \(f(x)\) on the uniform grid \(\{sh\}\) \((s=0,\pm1,\pm2,\ldots)\). In what follows,

\[ \omega_p(f^{(k)},h) = \sup_{|t|\le h} \left\{ \int_0^1 |f^{(k)}(x+t)-f^{(k)}(x)|^p\,dx \right\}^{1/p}, \]

where in the nonperiodic case the derivative \(f^{(k)}(x)\) outside the interval \([0,1]\) may be extended, for example, by zero.

Theorem 1. If the function \(f(x)\in W_p^k\), then for any \(p,q\ge1\), including the case \(k=1,\ 1\le p<2q(q+1)^{-1},\ p<q\), the inequality

\[ \inf \|f-S_{k,n}\|_{L_p(0,1)} \le Cn^{-k}\omega_p(f^{(k)},n^{-1}), \tag{2} \]

holds, where the infimum is taken over all splines \(S_{k,n}(x)\) of order \(k\) with a number of knots not exceeding \(n\), and the constant \(C\) depends only on \(k\).

Theorem 2. If the function \(f(x)\in W_p^k\) and \(m\ge k\), then

\[ \inf_{S_{m,n}} \|f-S_{m,n}\|_{L_p(0,1)} \le C(k,m)n^{-k}\|f^{(k)}\|_{L_p(0,1)} \qquad (p,q\ge1), \tag{3} \]

where \(S_{m,n}(x)\) is a spline of order \(m\) with a number of knots not exceeding \(n\).

For \(m=k-1\) inequality (3) was proved in \((^1)\).

Theorem 3. There exist splines \(S_{m,n}(x)\) and \(S_{k,n}(x)\) such that \((m \ge k)\)

\[ \|f^{(i)}-S_{m,n}^{(i)}\|_{L_p(0,1)} \le C(k,m)n^{-k+i}\|f^{(k)}\|_{L_p(0,1)} \quad (p,q \ge 1,\ 0 \le i \le k), \tag{4} \]

\[ \|f^{(i)}-S_{k,n}^{(i)}\|_{L_q(0,1)} \le C(k)n^{-k+i}\omega_p(f^{(k)},n^{-1}) \quad (p,q \ge 1,\ 0 \le i < k,\ k>1). \tag{5} \]

The example of the function \(f_n(x)\),
\[ f_n^{(k)}(x)=(x-x_s)^{m+1-k}n^{m+1-k}, \quad x_s \le x \le x_{s+1}, \]
\(x_s=s/n\) \((s=0,1,\ldots,n-1)\), shows that estimate (3) cannot be improved.

Let a sequence of meshes be given

\[ \Delta_s:\ 0=x_0^{(s)}<x_1^{(s)}<\cdots<x_{n_s}^{(s)}=1,\quad n_s=sm \quad (s=1,2,\ldots). \tag{6} \]

Theorem 4. If the quantities \((t_i=x_{im})\)

\[ R_s=\max_{0\le i\le s-1}\ \max_{t_i\le t\le t_{i+1}} \sum_{r=0}^{m-1} \frac{\psi_i(t)}{\psi_i'(x_{im+r})(t-x_{im+r})} \le \bar\beta<\infty, \tag{7} \]

\[ \psi_i(t)=(t-x_{im})(t-x_{im+1})\cdots(t-x_{i(m+1)}), \]

then there exists a spline \(S_{m,n_s}(x)\) with knots (6) such that

\[ \|f^{(i)}-S_{m,n_s}^{(i)}\|_{L_q(0,1)} \le C_1\|\Delta_s\|^{\gamma-i}\|f^{(k)}(x)\|_{L_p(0,1)}, \tag{8} \]

where

\[ \|\Delta_s\|=\max_{0\le i\le n_s-1}|x_{i+1}^{(s)}-x_i^{(s)}|, \quad \gamma=k\ (q\le p) \quad \text{and} \quad \gamma=k+q^{-1}-p^{-1}\ (q>p). \]

If \(m=k\), then \(\|f^{(k)}\|_{L_p(0,1)}\) may be replaced by \(\omega_p(f^{(k)},\|\Delta_s\|)\).

If the spline \(S_m(x,h)\) interpolates the function \(f(x)\in W^kL_p(-\infty,\infty)\) at the knots \(sh\) \((s=0,\pm1,\pm2,\ldots)\), then the following holds.

Theorem 5. For \(q \ge p\) and \(m>k\ge1\) the inequality

\[ \|f^{(i)}(x)-S_m^{(i)}(x,h)\|_{L_p(-\infty,\infty)} \le Ch^{k+1/q-1/p-i}\omega_p(f^{(k)},h) \quad (0\le i\le k-1), \tag{9} \]

holds, where \(C\) depends only on \(k\) and \(m\), and

\[ \omega_p(f^{(k)},h)= \sup_{|t|\le h} \|f^{(k)}(x+t)-f^{(k)}(x)\|_{L_p(-\infty,\infty)}. \tag{10} \]

In the proof of Theorems 1 and 2, a certain extremal set of knots is constructed (for each function its own)
\[ 0=x_0<x_1<\cdots<x_s=1, \]
and then, on each interval \([x_i,x_{i+1}]\), the spline \(S_{m,m,i}(x)\) is determined from the conditions

\[ f^{(\alpha)}(x_r)=S_{m,m,i}^{(\alpha)}(x_r),\quad \alpha=0,1,\ldots,k-1; \tag{11} \]

\[ S_{m,m,i}^{(\alpha)}(x_r)=0,\quad \alpha=k,\ldots,m-1\ (m>k);\ r=i,i+1. \]

Here the intermediate knots of the spline \(S_{m,m,i}(x)\) are chosen in the form
\[ x_i^{(l)}=x_i+\lambda_l(x_{i+1}-x_i), \]
where the numbers \(\lambda_l\) do not depend on \(i\) and satisfy the inequalities
\[ 0=\lambda_0<\lambda_1<\cdots<\lambda_m=1. \]

The proof of Theorem 1 reduces to computing the quantity

\[ \beta=\inf_{\{x_i\}}\max_{0\le i\le s-1}\beta_i, \tag{12} \]

\[ \beta_i=\Delta x_i^{kp-2+p/q} \int_0^{\Delta x_i} dt \int_0^{\Delta x_i} |f^{(k)}(t+u+x_i)-f^{(k)}(u+x_i)|^p\,du, \tag{13} \]

where \(\Delta x_i = x_{i+1}-x_i\). For \(kp-2+p/q \ge 0\), just as in \((^1)\), it is also proved that the infimum is attained when all \(\beta_i\) are equal. The proof of Theorem 2 proceeds analogously, only in the present case

\[ \beta_i=\Delta x_i^{kp-1+p/q}\int_0^{\Delta x_i}\left|f^{(k)}(u+x_i)\right|^p\,du. \tag{14} \]

Comparison of Theorems 1 and 2 with Theorem 4 shows that for \(q>p\) a special choice of knots ensures a better order of convergence.

In the proof of Theorem 4 the representation for \(S_m^{(m)}(x,h)\) found in \((^5)\) is used. For \(p=q=\infty\) this theorem was proved in \((^6)\).

REFERENCES

\(^1\) Yu. N. Subbotin, N. I. Chernykh, Matem. zametki, 7, no. 1, 31 (1970).
\(^2\) I. J. Ahlberg, E. N. Nilson, I. L. Walsh, The Theory of Splines and their Applications, N. Y.—London, 1967.
\(^3\) V. M. Tikhomirov, Matem. sbornik, 80 (122), 290 (1969).
\(^4\) I. J. Schoenberg, Quart. Appl. Math., 4, 45, 112 (1946).
\(^5\) Yu. N. Subbotin, Matem. zametki, 1, no. 1, 63 (1967).
\(^6\) Yu. N. Subbotin, Matem. zametki, 7, no. 1, 43 (1970).