V. F. Gaposhkin

THE LOCALIZATION PRINCIPLE AND SYSTEMS \(\{\varphi(nx)\}\)

(Presented by Academician P. S. Novikov on 29 XII 1961)

In the author’s paper \((^1)\) a class of systems of the form \(\{\varphi(nx)\}\), “close” in their properties to the trigonometric system, was singled out. This class is characterized by the following properties:

a)
\[ \varphi(x)=\sum_{k=1}^{\infty} b_k \sin kx,\quad \text{where}\quad \sum_{k=1}^{\infty} |b_k|\leq C_2<\infty; \]

b) the function
\[ \Phi(z)=\sum_{k=1}^{\infty}\frac{b_k}{k^z} \]
satisfies the inequalities
\[ 0<C_1\leq |\Phi(z)|\leq C_2\quad \text{for } \operatorname{Re} z>0. \]

When conditions a) and b) are fulfilled, the system \(\{\varphi(nx)\}_{n=1}^{\infty}\) is a complete minimal system in \(L'_p(-\pi,\pi)\), \(p\geq 1\) (the space of odd \(2\pi\)-periodic functions with integrable \(p\)-th power), and the adjoint system has the form
\[ \psi_n(x)=\sum_{d/n} B_d \sin \frac{n}{d}x \]
(the sum is taken over all integer divisors of the number \(n\)), where the numbers \(B_n\) are uniquely determined from the equations
\[ B_1b_1=1,\qquad \sum_{d/n} B_d b_{\frac{n}{d}}=0\quad (n>1). \tag{1} \]

The Fourier series of a function \(f(x)\in L'(-\pi,\pi)\) with respect to the system \(\{\varphi(nx)\}\) has the form
\[ f(x)\sim \sum_{n=1}^{\infty} \widetilde{a}_n \varphi(nx), \tag{2} \]
where
\[ \widetilde{a}_n=(f,\psi_n)=\sum_{d/n} B_d a_{\frac{n}{d}};\qquad a_n=\frac{2}{\pi}\int_0^\pi f(x)\sin nx\,dx, \]
\[ f(x)\sim \sum_{n=1}^{\infty} a_n \sin nx. \tag{3} \]

In paper \((^1)\) it was shown that, when conditions a) and b) are fulfilled, the series (2) and (3) possess many similar properties. For example, they simultaneously converge or diverge in the metric \(C'(-\pi,\pi)\); \(L'_p(-\pi,\pi)\) \((p\geq 1)\), simultaneously converge absolutely or fail to converge absolutely, etc. (see also papers \((^2)\), where a narrower class of systems was considered). It may be observed that the theorems of papers \((^1,^2)\) concern properties of the series (2) and (3) depending on the behavior of the odd function \(f(x)\) on the whole interval \((0,\pi)\). It is natural to ask whether any theorems on equiconvergence at individual points will hold for the series (2) and (3). It is known that convergence of the series (3) at a point \(x_0\) depends only on the properties of the function \(f(x)\) in some neighborhood of this point (Riemann’s localization principle for the trigonometric system). Therefore our question reduces to the following: is the localization principle valid for the systems under consideration? The following Theorems 1 and \(1'\) give a negative answer to this question.

We shall say, as usual, that for the system \(\{\varphi_k(x)\}\) the localization principle is valid for convergence (summability by method \(A\)) at a certain point-

point \(x_0\), if from the coincidence of two functions \(f_1(x)\) and \(f_2(x)\) in some neighborhood of the given point it follows that

\[ \lim_{n\to\infty}\{S_n(f_1;x_0)-S_n(f_2;x_0)\}=0, \]

where \(S_n(f_i;x)\) are the partial sums (or \(A\)-means) of the Fourier series of the functions \(f_i(x)\) \((i=1,2)\) with respect to the system \(\{\varphi_k(x)\}\), \(f_i\in L'(-\pi,\pi)\). We shall call systems possessing properties a) and b) \(T\)-systems.

Theorem 1. If for the \(T\)-system \(\{\varphi(nx)\}\) the localization principle for Abel’s summability method is valid at some point \(x_0\), \(x_0\ne \frac{p}{q}\pi\), then

\[ \varphi(x)=b_1\sin x,\qquad b_1\ne 0. \]

Here and below \(p\) and \(q\) are relatively prime integers, \(p<q\), \(q>1\).

From Theorem 1 there follows immediately:

Theorem \(1'\). If for the \(T\)-system \(\{\varphi(nx)\}\) the localization principle for convergence is valid at some point \(x_0\), \(x_0\ne \frac{p}{q}\pi\), then \(\varphi(x)=b_1\sin x\), \(b_1\ne 0\).

If we now consider the case when \(x_0=\frac{p}{q}\pi\), then the function \(\varphi(x)\) in the hypotheses of Theorem 1 (or Theorem \(1'\)) is no longer obliged to coincide (up to a factor) with \(\sin x\). Put

\[ \gamma_{l,\nu}=\sum_{plk\equiv \nu\pmod q} b_k\,\operatorname{sign}\left(\sin \frac{plk}{q}\pi\right), \qquad 1\le \nu\le q-1;\quad l=1,2,\ldots \]

Notice that \(|\gamma_{l,\nu}|=|\gamma_{sq+l,\nu}|\) for \(s\ge 1\).

Theorem 2. Let \(\{\varphi(nx)\}\) be a \(T\)-system. In order that, for this system, the localization principle be valid at some point \(x_0=\frac{p}{q}\pi\), it is necessary and sufficient that the relations

\[ B_l\gamma_{l,\nu}=0,\qquad 1\le \nu\le q-1;\quad \nu\ne p;\quad l=1,2,\ldots \tag{4} \]

be satisfied.

It is also easy to show that, for points \(x_0\) of the form \(\frac{p}{q}\pi\), the following modified localization principle is valid:

The convergence of the Fourier series of a function \(f(x)\) at the point \(x_0\) with respect to any \(T\)-system \(\{\varphi(nx)\}\) depends only on the behavior of the function \(f(x)\) in neighborhoods of the points \(x_0,2x_0,\ldots,(q-1)x_0\).

Let us make the following additional remarks.

1. For any point \(x_0=\frac{p}{q}\pi\) there exist \(T\)-systems \(\{\varphi(nx)\}\), \(\varphi(x)\ne b_1\sin x\), for which the localization principle is valid at this point (i.e. conditions (4) are satisfied). As such a system one may take the system generated by the function \(\varphi(x)=\sin x-\frac12\sin qx\). In this case \(B_1=1\),

\[ B_{q^n}=\frac{1}{2^n}\ (n\ge 1),\qquad B_k=0\ (k\ne q^n). \]

1. All the results carry over without difficulty to \(T\)-systems of the form \(\{1,\varphi(nx)\}\) in the case when the function \(\varphi(x)\) is even.

2. The results of the present note show that systems \(\{\varphi(nx)\}\) close to the system \(\{\sin nx\}\) in their “global” properties turn out to be far from it in their local properties. However, the possibility is not excluded that among systems \(\{\varphi(nx)\}\) which are not \(T\)-systems there may be some for which the localization principle is valid.

Received
26 XII 1961

REFERENCES

1. V. F. Gaposhkin, Matem. sborn., 51 (93), 2, 239 (1960).
2. K. F. Mayavko, DAN, 118, No. 1, 29 (1958).

* Theorem 1 remains valid if one considers the localization principle in the class of bounded functions, and Theorem \(1'\) in the class of continuous functions.