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MATHEMATICS
S. A. AGAKHANOV, G. I. NATANSON
THE GIBBS PHENOMENON
FOR CERTAIN SUMMATION PROCESSES OF FOURIER SERIES
(Presented by Academician L. V. Kantorovich, 25 XII 1964)
This note establishes sufficient conditions for the presence of the Gibbs phenomenon for certain triangular methods of summation of trigonometric Fourier series.
- Introduce the notation:
\[ \lambda_k^{(n)}=\varphi(k/n)+\alpha_{k,n}, \tag{1} \]
where \(n=1,2,\ldots;\ k=0,1,\ldots,n;\) the function \(\varphi(t)\) is defined on \([0,1]\), \(\varphi(0)=1;\)
\[ S_n(\lambda,t)=\sum_{k=1}^{n}\lambda_k^{(n)}\frac{\sin kt}{k}, \qquad S_n(\varphi,t)=\sum_{k=1}^{n}\varphi\left(\frac{k}{n}\right)\frac{\sin kt}{k}; \]
\[ I_{\varphi}(y)=\int_{0}^{1}\varphi(t)\frac{\sin yt}{t}\,dt; \qquad \psi(t)=\frac{\varphi(t)-1}{t}; \qquad \chi(t)=\frac{\varphi(t)}{t}. \]
Lemma. If \(\varphi(t)\) is \(R\)-integrable,
\[ \sum_{k=1}^{n}\left|\varphi\left(\frac{k-1}{n}\right)-\varphi\left(\frac{k}{n}\right)\right|\frac{n-k}{n} =O(1), \quad \alpha_{k,n}=o\bigl((\ln n)^{-1}\bigr), \quad \sup_{0\le y<\infty} I_{\varphi}(y)>\pi/2, \]
then, in summing the series
\[ \sum_{k=1}^{\infty}\frac{\sin kt}{k} \]
with multipliers (1), the Gibbs phenomenon occurs at the point \(t=0\), i.e.
\[ \overline{\lim_{\substack{n\to\infty\\ t\to 0+}}} S_n(\lambda,t)>\pi/2. \tag{2} \]
Proof. First of all, note that under our conditions
\[ S_n(\varphi,t)-S_n(\lambda,t)=o(1) \]
uniformly with respect to \(t\). Applying Abel’s transformation, we find
\[ S_n(\varphi,t) = \varphi\left(\frac{1}{n}\right)\sum_{k=1}^{n}\frac{\sin kt}{k} - \sum_{k=2}^{n} \left[ \varphi\left(\frac{k-1}{n}\right)-\varphi\left(\frac{k}{n}\right) \right] \sum_{\nu=k}^{n}\frac{\sin \nu t}{\nu}. \]
Let \(nt_n\to y\), where \(0\le y<\infty\). We have
\[ S_n(\varphi,t_n)-S_n\left(\varphi,\frac{y}{n}\right) = \varphi\left(\frac{1}{n}\right) \sum_{k=1}^{n}\frac{\sin kt_n-\sin ky/n}{k} - \]
\[ - \sum_{k=2}^{n} \left[ \varphi\left(\frac{k-1}{n}\right)-\varphi\left(\frac{k}{n}\right) \right] \sum_{\nu=k}^{n}\frac{\sin \nu t_n-\sin \nu y/n}{\nu}. \]
But
\[ \sum_{\nu=k}^{n}\nu^{-1}\left(\sin \nu t_n-\sin \nu\frac{y}{n}\right) = \sum_{\nu=k}^{n}\left(t_n-\frac{y}{n}\right)\cos\theta_\nu = \frac{n-k}{n}\,O(nt_n-y). \]
Therefore \(S_n(\varphi,t_n)-S_n\left(\varphi,\frac{y}{n}\right)=o(1)\). On the other hand, \(S_n(\varphi,y/n)\to I_\varphi(y)\). Hence \(S_n(\varphi,t_n)\to I_\varphi(y)\), whence
\[ I_\varphi(y)\leq \varlimsup_{\substack{n\to\infty\\ t\to 0+}} S_n(\varphi,t), \]
which implies (2).
Remark. An analogous assertion, but under somewhat different conditions, was established by O. Szasz \((^1)\).
Theorem 1. Let \(a_{k,n}=o((\ln n)^{-1})\), \(\sum_{k=0}^{n}|a_{k,n}|=O(1)\). If for some integer \(m\geq 0\): a) the function \(\psi^{(m)}(t)\) is absolutely continuous on \([0,1]\); b) \(\varphi^{(2p+1)}(0)=\varphi^{(q)}(1)=0\) \((p=0,1,\ldots,[(m-1)/2],\ q=0,1,\ldots,m-1)\); c) for odd \(m\), \(\varphi^{(m)}(1)\ne 0\), and for even \(m\), \((-1)^{m/2+1}\varphi^{(m+1)}(0)<(m+1)|\varphi^{(m)}(1)|\), then for the process generated by the multipliers (1), at the points of discontinuity of an arbitrary function of bounded variation the Gibbs phenomenon occurs.
Remark 1. Obviously, it is sufficient to require that \(a_{k,n}=O(n^{-1})\).
Remark 2. The existence of \(\varphi^{(m+1)}(0)\) follows from the equality
\[ \varphi^{(m+1)}(0)=(m+1)\psi^{(m)}(0). \]
Proof. From the absolute continuity of \(\psi(t)\) and the restrictions imposed on \(a_{k,n}\), there follows the regularity of the process under consideration. Therefore it is sufficient to study the behavior of the sums \(S_n(\lambda,t)\).
Since
\[ I_\varphi(y)=\frac{\pi}{2}-\int_{1}^{\infty}\frac{\sin yt}{t}\,dt+\int_{0}^{1}\psi(t)\sin yt\,dt, \]
then, integrating by parts \(m+1\) times, we obtain
\[ \begin{aligned} I_\varphi(y) &=\frac{\pi}{2} +\sum_{i=0}^{[m/2]} \frac{(-1)^i\big[(2i+1)^{-1}\varphi^{(2i+1)}(0)-\chi^{(2i)}(1)\cos y\big]}{y^{2i+1}} \\ &\quad -\sum_{i=1}^{[(m+1)/2]} \frac{(-1)^i\chi^{(2i-1)}(1)\sin y}{y^{2i}} -\frac{(m+1)!}{y^{m+1}} \int_{1}^{\infty} \frac{\sin\left(yt-\frac{m+1}{2}\pi\right)}{t^{m+2}}\,dt \\ &\quad +\frac{(-1)^{m+1}}{y^{m+1}} \int_{0}^{1}\psi^{(m+1)}(t)\sin\left(yt-\frac{m+1}{2}\pi\right)\,dt . \end{aligned} \tag{3} \]
But
\[ \int_{1}^{\infty}t^{-m-2}\sin\left(yt-\frac{m+1}{2}\pi\right)\,dt=O(y^{-1}) \]
and, by the Riemann–Lebesgue theorem,
\[ \int_{0}^{1}\psi^{(m+1)}(t)\sin\left(yt-\frac{m+1}{2}\pi\right)\,dt=o(1)\qquad (y\to\infty). \]
Moreover,
\[ \chi^{(q)}(1)=\varphi^{(q)}(1)+\sum_{\nu=0}^{q-1}a_\nu\varphi^{(\nu)}(1). \]
Thus, for odd \(m\),
\[ I_\varphi(y)=\frac{\pi}{2} -\frac{(-1)^{(m+1)/2}\varphi^{(m)}(1)\sin y}{y^{m+1}} +o(y^{-m-1}), \]
and for even \(m\)
\[ I_\varphi(y)=\frac{\pi}{2}+ \frac{(-1)^{m/2}\left[(m+1)^{-1}\varphi^{(m+1)}(0)-\varphi^{(m)}(1)\cos y\right]}{y^{m+1}} +O\left(y^{-m-1}\right). \]
Hence, from condition c) of the theorem, it follows that there exist such \(y\) that \(I_\varphi(y)>\pi/2\). Using the lemma, we obtain what was required.
Theorem 2. If \(\varphi(t)\) is an even function, analytic in the closed unit disk, \(\varphi(0)=1\), and, as above, \(a_{k,n}=o((\ln n)^{-1})\),
\[ \sum_{k=0}^{n}|a_{k,n}|=O(1), \]
then for the process generated by the multipliers (1), the Gibbs phenomenon occurs.
Proof. In the present case \(\varphi^{(2p+1)}(0)=0\) for any \(p\), and there is an \(m\) such that \(\varphi^{(q)}(1)=0\) for \(0\le q\le m-1\) and \(\varphi^{(m)}(1)\ne0\). Since \(\psi^{(m)}(t)\) is trivially absolutely continuous on \([0,1]\), all the conditions of Theorem 1 are satisfied.
Example 1. Partial sums of the Fourier series: \(\lambda_k^{(n)}=1\), \(\varphi(t)\equiv1\). Here the conditions of Theorem 2 are satisfied.
Example 2. The Bernstein–Rogozinsky summation method:
\[ \lambda_k^{(n)}=\cos^p\frac{k\pi}{2n+1} =\cos^p\frac{k\pi}{2n}+O(n^{-1}),\qquad \varphi(t)=\cos\frac{p\pi t}{2}. \]
The Gibbs phenomenon exists for any integer \(p\) on the basis of Theorem 2. This is a result of F. I. Kharshiladze \((^2)\).
Example 3. The Jackson–Vallée-Poussin method (see, for example, \((^3)\), p. 133). The kernel of this method is nonnegative. Hence there is no Gibbs phenomenon. Let us see what Theorem 1 gives. We may take
\[ \varphi(t)= \begin{cases} 1-6t^2+6t^3, & 0\le t\le \tfrac12,\\ 2(1-t)^3, & \tfrac12\le t\le1. \end{cases} \]
Then \(\varphi(1)=\varphi'(1)=\varphi''(1)=\varphi'(0)=0\), \(\varphi'''(0)=36\). Taking \(m=2\), we see that \(\psi''(t)\) is absolutely continuous, but
\[ (-1)^{m/2+1}\varphi^{(m+1)}(0)=36>0=\varphi^{(m)}(1), \]
i.e., condition c) is violated.
Example 4. The method of L. P. Lebedeva \((^4)\): \(\lambda_k^{(n)}=1-e^{k/n-n/k}\). Here \(\varphi(t)=1-e^{t-1/t}\), \(\varphi'(0)=\varphi(1)=0\), \(\varphi'(1)=-2\). The presence of the Gibbs phenomenon follows from Theorem 1 with \(m=1\).
Example 5. Truncated means of partial sums of the Fourier series:
\[ \lambda_k^{(n)}= \begin{cases} 1, & 0\le k/n<\theta,\\ (n-k)/m, & \theta\le k/n\le1, \end{cases} \]
\(0<\theta<1,\ m=n-[\theta n]\). As the function \(\varphi\) one should take
\[ \varphi(t)= \begin{cases} 1, & 0\le t<\theta,\\ \dfrac{1-t}{1-\theta}, & \theta\le t\le1. \end{cases} \]
Theorem 1 is not applicable either for \(m=0\), since \(\varphi'(0)=\varphi(1)=0\), or for \(m\ge1\), since the function \(\psi'(t)\) is discontinuous. However, using formula (3) for \(m=0\), we obtain
\[ I_\varphi(y)=\frac{\pi}{2} +\frac1y\int_1^\infty \frac{\cos yt}{t^2}\,dt -\frac1y\int_\theta^1 \frac{\theta}{1-\theta}\frac{\cos yt}{t^2}\,dt = \]
\[ =\frac{\pi}{2} +\frac{\sin\theta y-\theta\sin y}{\theta(1-\theta)y^2} +O(y^{-3}). \]
Since \(\sin \theta y > \theta \sin y\) for \(y = (4\nu + 1)\pi/2\theta\), the Gibbs phenomenon therefore also exists for this process for any \(\theta \in (0,1)\), as was established by another method in work \((^5)\) by one of the authors of the present article.
Example 6. In the work of A. Kh. Turetskii \((^6)\) the following special case of the Voronoi summability method is considered:
\[ \lambda_k^{(n)} = (n+k+2)(n-k+1)/(n+2)(n+1). \]
It is clear that \(\lambda_k^{(n)} = 1 - k^2/n^2 + O(n^{-1})\), i.e. \(\varphi(t)=1-t^2\). The Gibbs phenomenon takes place by Theorem 2.
Example 7. The means of M. Riesz: \(\varphi(t)=(1-t^\lambda)^k\). We restrict ourselves to the case of natural \(\lambda\) and \(k\). If \(\lambda\) is even, then the existence of the Gibbs phenomenon follows from Theorem 2. This fact for \(\lambda=2\) is given in \((^1)\). Let now \(\lambda=2l+1\). Then the function \(\psi(t)\) is a polynomial, \(\varphi^{(2p+1)}(0)=0\) for \(0 \le p \le l-1\), \(\varphi^{(2l+1)}(0)=-k(2l+1)!\), \(\varphi^{(q)}(1)=0\) for \(0 \le q \le k-1\), \(\varphi^{(k)}(1)=(-1)^k k!(2l+1)^k\). If \(k \le 2l\), then Theorem 1 with \(m=k\) gives the existence of the Gibbs phenomenon. If, however, \(k>2l\), then condition c) of Theorem 1 (\(m=2l\)) becomes the inequality \((-1)^{l+2}k(2l+1)!<0\), which is fulfilled for odd \(l\). Thus, if \(\lambda=2l+1\), then for \(k \le 2l\) the Gibbs phenomenon exists for all \(l\), while for \(k>2l\) it exists for odd \(l\). In the case of even \(l<k/2\), Theorem 1 gives no answer. We note that for \(l=0\) and any \(k \ge 1\) the Gibbs phenomenon is absent, since the corresponding means are the \(k\)-th iterations of the Fejér means, as indicated in \((^7)\).
Leningrad State
Pedagogical Institute
Received
16 XII 1964
REFERENCES
\(^1\) O. Szasz. Publ. Inst. Math. Acad. Serbe Sci., 4, 135 (1952).
\(^2\) F. I. Kharshiladze, DAN, 101, 425 (1955).
\(^3\) N. I. Akhiezer, Lectures on Approximation Theory, 1947.
\(^4\) L. P. Lebedeva, DAN, 142, 530 (1962).
\(^5\) G. I. Natanson, Investigations on Contemporary Problems of Constructive Function Theory, 1961, p. 206.
\(^6\) A. Kh. Turetskii, Izv. AN SSSR, Ser. Mat., 25, 411 (1961).
\(^7\) I. P. Natanson, Izv. Vyssh. Uchebn. Zaved., Matematika, 3, 93 (1964).