Abstract Generated abstract
This note studies approximation of bounded periodic functions in a Hausdorff-type metric defined through supplementary graphs, allowing treatment of functions with jumps under a modulus of nonmonotonicity. It establishes a general estimate for convolution approximants with nonnegative even kernels, relating the Hausdorff distance to the modulus of nonmonotonicity and the kernel tail. The result is applied to Fejér sums, Jackson polynomials, and Vallée-Poussin integrals, yielding explicit convergence rates under several assumptions on the modulus, including order estimates that are stated to be unimprovable in one case. A further theorem gives an analogous bound for Poisson means of the Fourier series and implies convergence in the Hausdorff-type metric as the Poisson parameter tends to one.
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MATHEMATICS
BLAGOVEST SENDOV
LINEAR METHODS OF APPROXIMATION OF PERIODIC FUNCTIONS WITH RESPECT TO A METRIC OF HAUSDORFF TYPE
(Presented by Academician A. N. Kolmogorov, 10 VIII 1964)
In this note some estimates are given for the Hausdorff distance for certain classical approximation apparatuses, such as Fejér sums, Jackson polynomials, and Vallée-Poussin integrals.
- At the beginning we define the notion of the modulus of nonmonotonicity \(\mu_f(\delta)\) of a function \(f(x)\) as follows:
\[ \mu_f(\delta)= \sup_{|x_1-x_2|\le \delta}\left\{ \sup_{x_1\le x\le x_2} \bigl[\,|f(x_1)-f(x)|+|f(x_2)-f(x)|\,\bigr] -|f(x_1)-f(x_2)| \right\}. \]
The modulus of nonmonotonicity \(\mu(\delta)\), with respect to the Hausdorff distance, plays a role analogous to that of the modulus of continuity with respect to the uniform distance.
- Let \(B_{2\pi}\) denote the class of \(2\pi\)-periodic functions \(f(x)\) for which: a) \(\lim_{\delta\to+0}\mu_f(\delta)=0\), and b) \(|f(x)|\le B\).
It is not difficult to prove that if \(f(x)\in B_{2\pi}\), then for every \(x\) there exist \(f(x-0)\) and \(f(x+0)\), and \(f(x)\) lies between \(f(x-0)\) and \(f(x+0)\).
- Let \(f(x)\) be a given function. Denote by \(\bar f\) a point set; we shall call it the supplementary graph of \(f(x)\), which is obtained as the intersection of all closed point sets \(F\) in the plane containing the graph of the function \(f(x)\) and convex with respect to the \(y\)-axis (i.e., every straight line parallel to the \(y\)-axis intersects \(F\) in a point or in a segment).
It is not difficult to see that if \(f(x)\in B_{2\pi}\), then \(\bar f\) consists of all points \(X(x,y)\) of the plane for which \(y\) lies between \(f(x-0)\) and \(f(x+0)\), while if \(f(x)\) is a continuous function, then \(\bar f\) coincides with the graph of the function \(f(x)\).
As in \((^{1-3})\), we define the distance \(r(f,g)\) between two functions \(f(x)\) and \(g(x)\) as follows:
\[ r(f,g)=\max\left\{ \max_{X\in \bar f}\min_{Y\in \bar g}\|X-Y\|_0,\, \max_{X\in \bar g}\min_{Y\in \bar f}\|X-Y\|_0 \right\}, \]
where
\[ \|X-Y\|_0=\|X(x_1,y_1)-Y(x_2,y_2)\|_0 =\max\bigl[\,|x_1-x_2|,\ |y_1-y_2|\,\bigr]. \]
Such a distance between point sets was considered by F. Hausdorff \((^4)\). Some properties of the distance \(r(f,g)\) with respect to continuous functions can be found in \((^1)\).
In working with the distance \(r(f,g)\), the following simple lemma is useful.
Lemma 1. If: a) for every point \(X\in \bar f\) there exists a point \(Y\in \bar g\) such that \(\|X-Y\|_0\le \delta\); b) for every point \(X\in \bar g\) there exists a point \(Y\in \bar f\) such that \(\|X-Y\|_0\le \delta\), then \(r(f,g)\le \delta\).
- Let \(K(t)\) be a function defined on the interval \([-\pi,\pi]\), with integrable square, for which: a) \(K(t)\geq 0\); b)
\[ \int_{-\pi}^{\pi} K(t)\,dt=1 \]
and c) \(K(t)=K(-t)\). If \(f(x)\in B_{2\pi}\), then by \(\varphi(t)=\varphi_f(t)\) we denote the function
\[ \varphi(x)=\int_{-\pi}^{\pi} f(x+t)K(t)\,dt . \tag{1} \]
Theorem 1. If \(f(x)\in B_{2\pi}\) and \(\varphi(x)\) is defined by equality (1), then
\[ r(f,g)\leq \inf_{\delta\geq 0}\max\left[\delta,\; \frac{1}{2}\mu(4\delta)+4B\int_{\delta}^{\pi}K(t)\,dt\right], \]
where \(\mu(\delta)\) is the modulus of nonmonotonicity of \(f(x)\).
The proof of the theorem, according to Lemma 1, follows from the following assertions.
If \(f(x)\in B_{2\pi}\) and has modulus of nonmonotonicity \(\mu(\delta)\), \(\varphi(x)\) is defined by equality (1), and \(\delta\geq 0\), then:
1) For each point \(X(x_1,y_1)\in \bar f\) there exists a point \(Y(x_2,y_2)\in \bar\varphi\) such that
\[ |x_1-x_2|\leq \delta,\qquad |y_1-y_2|\leq \frac{1}{2}\mu(4\delta)+2B\int_{\delta}^{\pi}K(t)\,dt . \]
2) For each point \(X(x_1,y_1)\in \bar\varphi\) there exists a point \(Y(x_2,y_2)\in \bar f\) such that
\[ |x_1-x_2|\leq \delta,\qquad |y_1-y_2|\leq 4B\int_{\delta}^{\pi}K(t)\,dt . \]
The latter assertions are obtained on the basis of the following property of functions from \(B_{2\pi}\).
Lemma 2. Let \(f(x)\in B_{2\pi}\) and have modulus of nonmonotonicity \(\mu(\delta)\), \(x_0=\frac{1}{2}(x_1+x_2)\), \(x_1<x_2\), and let \(y_0\) be an arbitrary number between \(f(x_0-0)\) and \(f(x_0+0)\). Then the inequality
\[ f(x)\geq y_0-\frac{1}{2}\mu(|x_1-x_2|) \]
holds for all \(x\in [x_1,x_0]\) or for all \(x\in [x_0,x_2]\), and the inequality
\[ f(x)\leq y_0+\frac{1}{2}\mu(|x_1-x_2|) \]
holds for all \(x\in [x_1,x_0]\) or for all \(x\in [x_0,x_2]\).
- We give some consequences of Theorem 1.
Denote the \(n\)-th Fejér sum by
\[ \sigma_n(x)=\frac{1}{2n\pi}\int_{-\pi}^{\pi} f(t)\left(\frac{\sin n(t-x)/2}{\sin (t-x)/2}\right)^2\,dt = \frac{1}{2n\pi}\int_{-\pi}^{\pi} f(x+t)\left(\frac{\sin nt/2}{\sin t/2}\right)^2\,dt, \]
the \(n\)-th Jackson polynomial by
\[ U_n(x)=\frac{3}{2\pi n(2n^2+1)}\int_{-\pi}^{\pi} f(x+t)\left(\frac{\sin nt/2}{\sin t/2}\right)^4\,dt; \]
the \(n\)-th Vallée-Poussin integral for a function \(f(x)\in B_{2\pi}\) with modulus of nonmonotonicity \(\mu(\delta)\) by
\[ V_n(x)=\frac{(2n)!!}{(2n-1)!!}\,\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x+t)\cos^{2n}\frac{t}{2}\,dt . \]
From Theorem 1 the following estimates are obtained:
1)
\[
r(f,\sigma_n)\leqslant \max\left[\delta,\frac12\mu(4\delta)+\frac{4B}{n\delta}\right],
\]
\[ r(f,U_n)\leqslant \max\left[\delta,\frac12\mu(4\delta)+\frac{10\pi^2B}{(n\delta)^3}\right], \]
\[ r(f,V_n)\leqslant \max\left[\delta,\frac12\mu(4\delta)+4B\sqrt{\pi n}\left(1-\frac{\delta^2}{\pi^2}\right)^{2n}\right]; \]
2) if \(\mu(\delta)\geqslant \delta\),
\[ r(f,\sigma_n)\leqslant (1+16B)\mu(n^{-1/2}), \]
\[ r(f,U_n)\leqslant (1+10\pi^2B)\mu(n^{-3/4}), \]
\[ r(f,V_n)\leqslant \mu(4\pi\sqrt{\ln n}/n)\quad \text{for } n\geqslant e^{8B^2/\pi}; \]
3) if \(\mu(\delta)\leqslant \delta\),
\[ r(f,\sigma_n)\leqslant 4\sqrt{2B}\,n^{-1/2}, \]
\[ r(f,U_n)\leqslant 12\sqrt[4]{Bn^{-3/4}}, \]
\[ r(f,V_n)\leqslant 2\pi\sqrt{\ln n}/n\quad \text{for } n\geqslant e^{64B^2/\pi}; \]
4) if \(\mu(\delta)\leqslant k\delta^\alpha\), for \(0<\alpha\leqslant 1\) and \(k>0\),
\[ r(f,\sigma_n)\leqslant 4\sqrt{2kB}\,n^{-\alpha/(1+\alpha)}+O(n^{-1/(1+\alpha)}), \]
\[ r(f,U_n)\leqslant cn^{-3\alpha/(3+\alpha)},\quad \text{where } c=\max[1,2k+10\pi^2B]. \]
It can be proved that the inequalities in item 4) cannot be improved in the sense of order.
Using the Poisson kernel, one can obtain the following theorem.
Theorem 2. Let \(f\in B_{2\pi}\) and
\[ f(x)\sim \frac{a_0}{2}+\sum_{m=1}^{\infty}(a_m\cos mx+b_m\sin mx) \]
be its Fourier series. Denote
\[ f_r(x)=\frac{a_0}{2}+\sum_{m=1}^{\infty}r^m(a_m\cos mx+b_m\sin mx); \]
then
\[ r(f,f_r)\leqslant \inf_{\delta\geqslant 0}\max\left[\delta,\frac12\mu(4\delta)+\frac{2B(1-r^2)}{r^2\delta}\right] \]
and, consequently,
\[
\lim_{r\to 1-0} r(f,f_r)=0.
\]
Mathematical Institute
with Computing Center
Sofia, Bulgaria
Received
1 VI 1964
REFERENCES
- Bl. Sendov, B. Penkov, Izv. na Mat. inst. pri BAN, 6, 27 (1962).
- Bl. Sendov, God. na Sof. univ., fiz.-mat. fak., 55, book 1, 1 (1960—1961).
- Bl. Sendov, God. na Sof. univ., fiz.-mat. fak., 56, book 1, 195 (1961—1962).
- F. Hausdorff, Set Theory, Moscow, 1936.