Mathematics

L. P. Lebedev

An Effective Method for Constructing Best Methods of Approximation

(Presented by Academician A. N. Kolmogorov, 28 VIII 1961)

By virtue of D. Jackson’s theorems on the influence of the structural properties of a function \(f(x)\) on the order of its approximation by trigonometric polynomials, we have

\[ \mathcal{E}U_n\bigl(KW^{(r)}H^{(\alpha)}, x\bigr) = \sup_{f\in KW^{(r)}H^{(\alpha)}} \bigl| f(x)-U_n(f;x)\bigr| = O\left(\frac{1}{n^{r+\alpha}}\right), \tag{1} \]

where \(KW^{(r)}H^{(\alpha)}\) \((r=0,1,2,\ldots,\ 0\le \alpha<1)\) is the class of functions \(f(x)\) of period \(2\pi\) having a derivative of order \(r\) satisfying a Lipschitz condition of degree \(\alpha\) with constant \(K\), and \(U_n(f;x)\) is a polynomial method of best approximation of order not exceeding \(n\).

Of interest is the problem of actually constructing a sequence of trigonometric polynomials \(U_n(f;x)\) realizing relation (1).

All concrete methods of summation of Fourier series known up to now of the form

\[ U_n(f;x;\lambda) = \frac{a_0}{2}\lambda^{(n)}_0 + \sum_{k=1}^{n} \lambda^{(n)}_k\bigl(a_k\cos kx+b_k\sin kx\bigr), \tag{2} \]

where \(a_k\) and \(b_k\) are the Fourier coefficients of the function \(f(x)\in C_{2\pi}\), for example the Fejér sums
\[ \left(\lambda^{(n)}_k=1-\frac{k}{n}\right), \]
the de la Vallée Poussin sums
\[ \left(\lambda^{(n)}_k=\frac{(n!)^2}{(n-k)!(n+k)!}\right), \]
the Bernstein–Rogozinskii sums
\[ \left(\lambda^{(n)}_k=\cos\frac{k\pi}{2n+1}\right), \]
their numerous generalizations (including Zygmund’s method:
\[ \lambda^{(n)}_k=1-\left(\frac{k}{n+1}\right)^p,\quad p=1,2,\ldots \]),
as well as all linear positive operators, are not a solution of this problem. Thus, the order of approximation of functions \(f(x)\in C_{2\pi}\) by the Fejér and de la Vallée Poussin operators cannot be better than \(1/n\), by the Bernstein–Rogozinskii polynomials and by all linear positive operators not better than \(1/n^2\), and by the Zygmund sums not better than \(1/n^p\); that is, all these modifications of Fourier sums have the common defect that the order of approximation effected by them on the classes \(KW^{(r)}H^{(\alpha)}\) is worse than the best one the better the function is. Let us note that approximation of functions of the classes \(KW^{(r)}H^{(\alpha)}\) by the Fourier sums themselves does not have such a defect, since it has, as is known, order \(\ln n/n^{r+\alpha}\).

This fact is apparently connected with the circumstance that whereas for all the methods mentioned

\[ \lim_{n\to\infty}\bigl(1-\lambda^{(n)}_k\bigr)n^m\ne 0, \quad m\ge m_0, \]

for the partial sums of the Fourier series we have

\[ \lim_{n\to\infty}\bigl(1-\lambda^{(n)}_k\bigr)n^m=0, \quad m=1,2,\ldots,\quad k=0,1,2,\ldots . \]

Thus, methods of summation of Fourier series of type (2) have hitherto been constructed in such a way that the removal of the defect of the Fourier partial sums (the absence of their uniform convergence on the whole class of continuous \(2\pi\)-periodic functions) was carried out simultaneously with the loss of a valuable quality of this approximation apparatus—its good approximative properties on classes of differentiable functions.

In the present note a simple method is described for constructing polynomials of type (2), uniformly convergent on the whole class of continuous \(2\pi\)-periodic functions, and realizing relation (1). In the course of the discussion this method appears as a special case of more general considerations in approximation theory.

Definition. A class of methods of summation of Fourier series of type (2) will be called the class \(e_n(f;x;\lambda)\), if

\[ \lambda_k=\lambda_k^{(n)}=1-e^{\varphi(n,k)},\qquad k=1,2,\ldots,n;\qquad \lambda_0^{(n)}=1,\quad \lambda_{n+1}^{(n)}=0 \tag{3} \]

and on \([1,n]\) \(\varphi(n,x)\) is twice differentiable, \(\varphi'(n,x)\ge 0\), and \(\varphi''(n,x)\) is continuous.

It is easy to see that all classical methods (if, instead of the Valle-Poussin factors, one takes a construction close to them with differentiable function \(\lambda_k^{(n)}=e^{-k^2/n}\)) belong to the class \(e_n(f;x;\lambda)\). However, of greatest interest are the summation methods of the class \(e_n(f;x;\lambda)\) proper, i.e., those for which the simplest representation of the convergence factors is the form (3).

Theorem 1. For the relation

\[ e_n(f;x;\lambda)\xrightarrow[n\to\infty]{} f(x)\in C_{2\pi} \tag{4} \]

to hold uniformly with respect to real \(x\), it is necessary that the conditions

\[ \lim_{n\to\infty}\varphi(n,k)=-\infty,\qquad k=1,2,\ldots; \tag{5} \]

\[ \lambda_{n-m}^{(n)}\ln\frac{n}{m+1}=O(1),\qquad \frac{m}{n}\xrightarrow[n\to\infty]{}0 \tag{6} \]

be satisfied.

The proof of the necessity of condition (6) follows directly from the equality

\[ \sum_{k=1}^{n}\frac{\lambda_k^{(n)}}{n-k+1} = \lambda_n^{(n)}\ln n+ \sum_{k=1}^{n-1}\Delta_k\ln\frac{n}{n-k}+O(1), \]

\[ \Delta_k=\lambda_k^{(n)}-\lambda_{k+1}^{(n)}. \]

The following theorem is valid for any method (2) with a function \(\lambda_x^{(n)}\) twice differentiable on \([1,n]\) (\(\lambda_x''\) is continuous).

Theorem 2. Suppose it is known that the function \(\lambda_x''\) can have only a finite number of real zeros in the interval \([1,n]\). Then, for relation (4) to hold, it is sufficient that conditions (5), (6) be satisfied with \(m=0\), and

\[ x\lambda_x'=O(1),\qquad x\le n. \tag{7} \]

If \(\lambda_x''\) does not change sign in the interval \([1,n]\) (the methods of Fejér, Zygmund, Bernstein—Rogosinski), then it is sufficient to require the fulfil-

condition (7) only for \(x=n\). If the roots \(x_i\) \((i=1,2,\ldots,i_0)\) of the second derivative \(\lambda_x''\) are known, then (7) is checked only for \(x=x_i\) and \(x=n\); if, moreover, \(\lambda_x'\leqslant 0\) on \([1,n]\) and \(\lambda_x''\) is nonnegative on \([x_{i_0},n]\) (the Vallée-Poussin method in the interpretation given above), then requirement (7) need be satisfied only for \(x=x_i\).

Let us add also that, if \(\Delta_k^2\geqslant 0\), \(k=1,2,\ldots,n-2\), and \(\lambda_x'\leqslant 0\) \((\varphi'(n,x)\geqslant 0)\) on \([1,n]\), then for the fulfillment of relation (4) it is necessary and sufficient that conditions (5) and (6) be observed for \(m=0\).

The proof of all these recommendations is easily obtained from the relations

\[ \int_0^\pi \left|\frac12+\sum_{k=1}^n \lambda_k^{(n)}\cos kt\right|\,dt \leqslant \frac2\pi \left|\lambda_n^{(n)}\right|\ln n +\frac\pi2\sum_{k=0}^{n-2}(k+1)\left|\Delta_k^2\right| +\frac\pi2 n\left|\Delta_{n-1}\right|, \]

\[ \Delta_k^2=\Delta_k-\Delta_{k+1}=\gamma\lambda_{k+\theta}'', \qquad k=1,2,\ldots,n-2,\qquad 0<\gamma<2,\qquad 0<\theta<2. \]

Theorem 3. Suppose that for the method \(e_n(f;x;\lambda)\) the following conditions are satisfied:

\[ \lambda_n^{(n)}\ln n=O(1), \tag{8} \]

\[ e^{\varphi(n,k)}=O\left[\left(\frac{k}{n_0}\right)^m\right], \tag{9} \]

or, in an equivalent form,

\[ \varphi(n,k)>O\left(\ln\frac{n_0}{k}\right),\qquad \varphi'(n,k)\leqslant O\left[\left(\frac{n_0}{k}\right)^{p_0}\frac1{k^{t_0}}\right], \tag{10} \]

where \(n_0\leqslant O(n)\); \(k<O(n_0)\); \(m\) is any positive number; \(p_0\) and \(t_0\geqslant 1\) are certain fixed real numbers. (It is easy to see that \(e^{\varphi(n,k)}=O(1)\), if \(n\geqslant k\geqslant O(n_0)\).)

Then, for any integer \(r\geqslant 0\) and \(0\leqslant\alpha<1\), the inequality

\[ \mathcal E_n = \begin{cases} \mathcal E_{e_n}\left(KW^{(r)}H^{(\alpha)},x\right)\\ \mathcal E_{e_n}\left(\widetilde{KW}^{(r)}H^{(\alpha)},x\right) \end{cases} \leqslant O\left(\frac{n^2}{n_0^{\,r+\alpha+2}}\right). \tag{11} \]

is valid.

The proof consists in estimating the sums entering into the formulation of A. F. Timan’s theorem ((\(^{1}\)), § 8.5.1); here one uses the equality

\[ \Delta^2\mu_k^{(n)}=\bar\gamma f''(k+\bar\theta), \qquad 0<\bar\gamma<2,\qquad 0<\bar\theta<2,\qquad k=1,2,\ldots,n-2; \]

\[ f(x)=\frac{e^{\varphi(n,x)}}{x^r}. \]

Estimate (11) is the more accurate, the better the method \(e_n(f;x;\lambda)\).

Corollary. If in the conditions of Theorem 3 \(n_0=O(n)\), then the method \(e_n(f;x;\lambda)\), on the classes \(KW^{(r)}H^{(\alpha)}\) and the conjugate classes, carries out uniform approximation of the best possible order, i.e. realizes relation (1).

Examples. Everywhere \(\lambda_0^{(n)}=1\), \(\lambda_{n+1}^{(n)}=0\), and the formulas given below are valid for \(k=1,2,\ldots,n\).

1. \(\lambda_k^{(1)}=1-e^{-e^{\,n-k}/n}\). Condition (6) is violated.

2. \(\lambda_k^{(2)}=1-e^{-e^{\,n-2k}}\). The conditions of Theorem 2 are violated.

1. \(\lambda_k^{(3)} = 1 - e^{-n/k^\gamma},\quad \gamma > 1.\) The conditions of Theorems 2 and 3 are satisfied for
   \[ n_0 = O\!\left(n^{1/\gamma}\right);\qquad \mathcal{E}_n \ll O\!\left(\frac{n^2}{n^{(r+\alpha+2)/\gamma}}\right). \]

2. \(\lambda_k^{(4)} = 1 - e^{-np/\ln n\cdot k^p},\quad p > 0.\) The conditions of Theorems 2 and 3 are satisfied for
   \[ n_0 = O\!\left[\frac{n}{(\ln n)^{1/p}}\right];\qquad \mathcal{E}_n \ll O\!\left[\frac{(\ln n)^{(r+\alpha+2)/p}}{n^{r+\alpha}}\right]. \]

3. \(\lambda_k^{(5)} = 1 - e^{-p\ln^\gamma(n/k)},\quad p > 0,\quad \gamma > 0\)—a generalization of the methods of Fejér \((p = 1,\ \gamma = 1)\) and Zygmund \((\gamma = 1)\). The conditions of Theorem 2 are satisfied and, if \(\gamma > 1\), those of Theorem 3 for \(n_0 = O(n)\); for \(\gamma > 1\)
   \[ \mathcal{E}_n = O\!\left(\frac{1}{n^{r+\alpha}}\right). \]

The last equality is also valid for the method:
\[ 6.\quad \lambda_k^{(6)} = 1 - e^{-n/k+k/n}. \]

Received
8 VIII 1951

REFERENCES

1. A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.