MATHEMATICS

N. N. GORBACH

ON THE SUMMATION OF FOURIER INTEGRALS OF FUNCTIONS OF SEVERAL VARIABLES BY THE RIESZ—BOCHNER METHOD

(Presented by Academician I. M. Vinogradov on 28 VI 1960)

Let \(KG^{(\alpha)}\) be the class of functions of two variables, summable over the entire plane,

\[ \iint\limits_{-\infty}^{\infty} |f(x,y)|\,dx\,dy<\infty \tag{1} \]

and such that the mean

\[ \bar f(t)=\frac{1}{2\pi}\int_{0}^{2\pi} f(x+t\cos\theta,\; y+t\sin\theta)\,d\theta \]

satisfies the condition

\[ \bar f(t)\in K\operatorname{Lip}\alpha,\quad 0\leq \alpha\leq 1. \]

Consider the Riesz—Bochner operator \(\left({}^{1}\right)\)

\[ S_R^{(\delta)}(f;\,x,y)=\frac{1}{2\pi}\iint\limits_{\nu\leq R} K_\delta\left(\frac{\nu}{R}\right)a(\alpha,\beta)e^{i(\alpha x+\beta y)}\,d\alpha\,d\beta, \quad \nu^2=\alpha^2+\beta^2, \]

where \(a(\alpha,\beta)\) is the Fourier transform of the function \(f(x,y)\);

\[ K_\delta(t)= \begin{cases} (1-t^2)^\delta, & \text{if } 0\leq t\leq 1,\\ 0, & \text{if } t>1, \end{cases} \qquad (\delta\geq 1/2) \]

as a method of approximating the function \(f(x,y)\in KG^{(\alpha)}\).

In this paper the quantity studied is

\[ \mathcal{E}_R^{(\delta)}\left(KG^{(\alpha)}\right) = \sup_{f\in KG^{(\alpha)}}\max_{(x,y)} \left|f(x,y)-S_R^{(\delta)}(f;\,x,y)\right|. \]

Analogous problems for Fourier sums were considered in the work of Chen Min-de and Chen Yun-xe \(\left({}^{3}\right)\).

Theorem 1. For the class of functions \(KG^{(\alpha)}\) \((\delta>1/2)\) the asymptotic equality holds

\[ \mathcal{E}_R^{(\delta)}\left(KG^{(\alpha)}\right) = O\left(\frac{1}{R^\alpha}\right). \]

Proof. Under the hypotheses of the theorem (see \(\left({}^{1}\right)\), pp. 176, 177) the representation

\[ S_R^{(\delta)}(f;\,x,y) = 2^\delta \Gamma(\delta+1)R^{1-\delta} \int_{0}^{\infty} \bar f(t)\, \frac{J_{\delta+1}(Rt)}{t^\delta}\,dt, \]

holds, where \(J_p(t)\) is the Bessel function of order \(p\).

The quantity \(\mathscr E_R^{(\delta)}(KG^{(\alpha)})\) does not depend on \(x\) and \(y\), and, since \(S_R^{(\delta)}(1;x,y)=1\), one may restrict oneself to considering functions for which \(f(0,0)=0\) (then \(\bar f(0)=0\)); moreover, without loss of generality one may take \(K=1\). We then have

\[ S_R^{(\delta)}(f;0,0)=2^\delta\Gamma(\delta+1)R^{1-\delta} \left[ \int_0^{\lambda(R)} \bar f_0(t)\frac{J_{\delta+1}(Rt)}{t^\delta}\,dt + \int_{\lambda(R)}^\infty \bar f_0(t)\frac{J_{\delta+1}(Rt)}{t^\delta}\,dt \right], \]

where

\[ \lambda(R)=O\left(\frac1R\right),\qquad \bar f_0(t)=\frac1{2\pi}\int_0^{2\pi} f(t\cos\theta,t\sin\theta)\,d\theta . \]

Since \(J_p(t)=O(t^p)\) as \(t\to 0\), it follows that

\[ \left| R^{1-\delta}\int_0^{\lambda(R)} \bar f_0(t)\frac{J_{\delta+1}(Rt)}{t^\delta}\,dt \right| = O\left(\frac1{R^\alpha}\right). \]

Using the asymptotic representation of the Bessel function

\[ J_p(c)=\sqrt{\frac2\pi}\, \frac{\cos(c-p\pi/2-\pi/4)}{c^{1/2}} + O\left(\frac1{c^{3/2}}\right) \qquad (c\to\infty), \]

and since \(\delta+{}^3\!/_{2}-\alpha>1\), we obtain

\[ \tag{2} S_R^{(\delta)}(f;0,0)= \]

\[ =2^\delta\Gamma(\delta+1)\sqrt{\frac2\pi}\,R^{1/2-\delta} \int_{\lambda(R)}^\infty \bar f_0(t) \frac{\cos\{Rt-(\delta+1)\pi/2-\pi/4\}}{t^{1/2+\delta}}\,dt + O\left(\frac1{R^\alpha}\right). \]

Let \(\gamma\) and \(m\), where \(0\le \gamma<\pi\) and \(m\) is a nonnegative integer, be such that

\[ \cos\left[Rt-\frac{\delta+1}{2}\pi-\frac\pi4\right] = (-1)^m\sin(Rt-\gamma), \]

\[ t_\mu=(\gamma+\mu\pi)/R,\qquad h=\pi/R. \]

Using the method of S. M. Nikol’skii \((^2)\) to estimate the first term on the right-hand side of (2), we obtain

\[ S_R^{(\delta)}(f;0,0) = 2^{\delta-1}\Gamma(\delta+1)R^{1/2-\delta} \sum_{\nu=1}^{\infty}(-1)^{m+\nu-1} \int_0^{h/2} \left[\varphi_\nu(t_\nu-u)-\varphi_\nu(t_\nu+u)\right]\times \]

\[ \times \left[ \frac1{(t_\nu-u)^{1/2+\delta}} + \frac1{(t_\nu+u)^{1/2+\delta}} \right]\sin Rt\,dt + O\left(\frac1{R^\alpha}\right), \tag{3} \]

where \(\varphi_\nu(t)=\bar f_0(t)-\bar f_0(t_\nu)\). Hence finally

\[ \mathscr E_R^{(\delta)}(KG^{(\alpha)})= O\left(\frac1{R^\alpha}\right). \]

Theorem 2. For the class of functions \(KG^{(\alpha)}\) \((\delta=1/2)\) the following asymptotic equality holds:
\[ \mathcal{E}_{R}^{(1/2)}\bigl(KG^{(\alpha)}\bigr) = \frac{K2^{\alpha+1}}{\pi}\, \frac{\ln R}{R^\alpha} \int_{0}^{\pi/2} u^\alpha \sin u\,du + O\left(\frac{1}{R^\alpha}\right). \]

Proof. a) Let \(0\leq \alpha<1\); then from (3) we obtain
\[ \left|S_{R}^{(1/2)}(f;0,0)\right| \leq \frac{2^{\alpha+1}}{\pi}\, \frac{\ln R}{R^\alpha} \int_{0}^{\pi/2} u^\alpha \sin u\,du + O\left(\frac{1}{R^\alpha}\right). \tag{4} \]

Consider a function of two variables, given in polar coordinates:
\[ f_R(t,\theta)= \begin{cases} 0, & 0\leq t\leq t_0,\quad t_{[R^2]}\leq t<\infty;\\ (-1)^{m+\nu}2^{\alpha+1}\pi\psi(\theta)(t-t_\nu)^\alpha, & t_\nu\leq t\leq t_{\nu+1/2};\\ (-1)^{m+\nu}2^{\alpha+1}\pi\psi(\theta)(t_{\nu+1}-t), & t_{\nu+1/2}\leq t\leq t_{\nu+1}, \end{cases} \]
\[ (\nu=0,1,2,\ldots,[R^2]-1), \]
where
\[ \int_{0}^{2\pi}\psi(\theta)\,d\theta=1. \]

It is not difficult to see that the function \(f_R(t,\theta)\) belongs to the class \(KG^{(\alpha)}\), and for it inequality (4) turns into an equality.

b) Let \(\alpha=1\). Using (1), we obtain
\[ \lim_{t\to\infty}\bar f_0(t)=0, \]
therefore
\[ \int_{\lambda(R)}^{\infty} \bar f(t)\, \frac{J_{1/2}(Rt)}{(Rt)^{1/2}}\,d(Rt) = \sqrt{\frac{2}{\pi}}\, \frac{1}{R} \int_{\lambda(R)}^{\infty} \bar f'_0(t)\, \frac{\sin Rt}{t}\,dt + O\left(\frac{1}{R}\right). \]

Thus we reduce the problem to the case \(\alpha=0\).

By \(KG^{(\alpha)}_{(r)}\) we denote the class of functions satisfying condition (1), for which \(\bar f(t)\) has an \(r\)-th \((r=0,1)\) derivative \(\bar f^{(r)}(t)\in K\operatorname{Lip}\alpha\), \(0\leq\alpha\leq1\).

Theorem \(2'\). For the class of functions \(KG^{(\alpha)}_{(r)}\) \((\delta=1/2)\) the following asymptotic equality holds:
\[ \mathcal{E}_{R}^{(1/2)}\bigl(KG^{(\alpha)}_{(r)}\bigr) = \frac{K2^{\alpha+1}}{\pi}\, \frac{\ln R}{R^{r+\alpha}} \int_{0}^{\pi/2} u^\alpha \sin u\,du + O\left(\frac{1}{R^{r+\alpha}}\right). \]

Analogous results also hold for functions of \(n\) variables, if one considers \(\delta\geq(n-1)/2\).

Remark. Denote by \(KW^{(\alpha)}\) the class of functions \(f(x)\), defined on \((0,\infty)\), for which the conditions
\[ \int_{0}^{\infty} x|f(x)|\,dx<\infty, \qquad f(x)\in K\operatorname{Lip}\alpha, \qquad 0\leq\alpha\leq1 \quad (0\leq x<\infty) \]
are satisfied.

We shall approximate functions of this class by means of the integral operators
\[ S_R^\delta(f) = 2^\delta\Gamma(\delta+1)R \int_{0}^{\infty} f(t+x)\, \frac{J_{\delta+1}(Rt)}{(Rt)^\delta}\,dt. \]

Corollary. For the class of functions \(KW^{(\alpha)}\) the following asymptotic equality holds:

\[ \sup_{f\in KW^{(\alpha)}}\ \sup_{0\le x<\infty}\left|f(x)-S_R^{(\delta)}(f)\right| = \begin{cases} O\!\left(\dfrac{1}{R^\alpha}\right), & \delta>\dfrac{1}{2},\\[1.2em] \dfrac{2^{\alpha+1}}{\pi}\,\dfrac{\ln R}{R^\alpha} \displaystyle\int_{0}^{\pi/2} u^\alpha \sin u\,du + O\!\left(\dfrac{1}{R^\alpha}\right), & \delta=\dfrac{1}{2}. \end{cases} \]

If \(f(x)\) has everywhere on \((0,\infty)\) a derivative \(f'(x)\in \operatorname{Lip}\alpha,\ 0\le \alpha\le 1\), then this result can be strengthened in the corresponding way.

L’vov State
University

Received
20 VI 1960

REFERENCES

\(^{1}\) S. Bochner, Trans. Am. Math. Soc., 40, 175 (1936).
\(^{2}\) S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 15 (1945).
\(^{3}\) RZh Matem., ref. 247 (1958).