MATHEMATICS

K. F. MALYAVKO

ON THE CONVERGENCE OF FOURIER SERIES WITH RESPECT TO SYSTEMS OF THE TYPE \(\{\varphi(nx)\}\), CLOSE TO THE TRIGONOMETRIC SYSTEM

(Presented by Academician A. N. Kolmogorov, 10 VII 1957)

Let \(L'_2[-\pi,\pi]\) be the space of odd complex-valued functions whose modulus squared is summable on \([-\pi,\pi]\). We shall regard every function as periodically extended with period \(2\pi\) to the entire real line. Given a function \(\varphi(x)\), construct the system of functions \(\{\varphi(nx)\}\), \(n=1,2,\ldots\). Questions of completeness of systems of this kind have been considered by many authors \((^{1-5})\).

Consider the Fourier series of the function \(\varphi(x)\) with respect to the trigonometric system \(\{\sin nx\}\)

\[ \varphi(x)\sim \sum_{k=1}^{\infty} b_k\sin kx . \tag{1} \]

If \(b_1\ne 0\), then it is not difficult to construct a system \(\{g_n(x)\}\) conjugate to the system \(\{\varphi(nx)\}\), i.e. a system for which \((\varphi(nx),g_k(x))=\delta_{nk}\). The system \(\{g_n(x)\}\) has the form

\[ g_n(x)=\sum_{k=1}^{n}\lambda_{nk}\sin kx . \]

For each function \(F(x)\in L'_2[-\pi,\pi]\) one can construct the Fourier series with respect to the system \(\{\varphi(nx)\}\)

\[ F(x)\sim \sum_{k=1}^{\infty} a_k\varphi(kx), \tag{2} \]

where the coefficients \(a_k\) are determined by the formulas

\[ a_k=\int_{-\pi}^{\pi} F(x)g_k(x)\,dx . \]

The problem of the present note is to find conditions that must be imposed on the function \(F(x)\) and on the system \(\{\varphi(nx)\}\) in order that the Fourier series of this function with respect to the system \(\{\varphi(nx)\}\) converge uniformly, everywhere, almost everywhere, or in the mean to \(F(x)\).

Consider the series with respect to the system \(\{\sin nx\}\) having the same coefficients as the series (2),

\[ \sum_{k=1}^{\infty} a_k\sin kx . \tag{3} \]

Suppose that the coefficients of the series (1) satisfy the condition

\[ \sum_{k=1}^{\infty}|b_k|<+\infty; \]

then

\[ \varphi(x)=\sum_{k=1}^{\infty} b_k\sin kx . \]

Denote the partial sum of the series (2) by \(S_n(x)\), and the partial sum of the series (3) by \(\tau_n(x)\). The relation holds

\[ S_n(x)=\sum_{k=1}^{\infty} b_k\tau_n(kx). \tag{4} \]

Suppose that the series (2) converges in mean to the function \(F(x)\), and the series (3) converges in mean to some function \(f(x)\in L_2'[-\pi,\pi]\). Then from equality (4) it follows that \(F(x)\) and \(f(x)\) are connected by the relation

\[ F(x)=\sum_{k=1}^{\infty} b_k f(kx). \tag{5} \]

Thus, the question of the mean convergence of the series (2) to \(F(x)\) can be studied by means of the functional equation (5), where \(F(x)\) is the given function; \(\{b_k\}\) are the given numbers and \(f(x)\) is the sought function.

In order to formulate conditions imposed on the coefficients \(\{b_k\}\) that are sufficient for solving equation (5), we shall use the well-known notion of a multiplicative function, as well as certain theorems obtained by V. Ya. Kozlov \((^5)\).

Let the integer \(n>0\) have the factorization into prime factors

\[ n=p_1^{\alpha_1(n)}\cdot p_2^{\alpha_2(n)}\cdots p_{k_n}^{\alpha_{k_n}(n)}, \tag{6} \]

where \(p_1,p_2,\ldots,p_i,\ldots\) are the prime numbers arranged in increasing order. To each prime number \(p_i\) we assign the complex variable \(z_i\). To the number \(n\) having the factorization (6), we assign the function

\[ \mu(n)=z_1^{\alpha_1(n)}\cdot z_2^{\alpha_2(n)}\cdots z_{k_n}^{\alpha_{k_n}(n)} \qquad (n=2,3,4,\ldots)\quad (\mu(1)=1). \]

The function \(\mu(n)\) is a multiplicative function of \(n\).

Consider the set of sequences of complex numbers \(z_1,z_2,\ldots,\ldots,z_i,\ldots\), satisfying the conditions: 1) \(|z_i|<1\); 2) \(\sum_{i=1}^{\infty}|z_i|^2<+\infty\).

We shall denote this set by \(G\). A sequence of complex numbers \(\{z_1,z_2,\ldots,z_i,\ldots\}\) will be called a point in the domain \(G\).

Definition \((^5)\). A kernel is a function of \(x\) and of a countable set of complex variables \(z_1,z_2,\ldots,z_i,\ldots\), defined in the following way:

\[ M(x;z_1,z_2,\ldots,z_i,\ldots)=\sum_{n=1}^{\infty}\mu(n)\sin nx. \tag{7} \]

Lemma (V. Ya. Kozlov \((^5)\)). For each fixed point of the domain \(G\), the series (7) converges in mean and the kernel \(M(x;z_1,z_2,\ldots,z_i,\ldots)\) belongs to \(L_2'[-\pi,\pi]\).

Definition \((^5)\). A function of a countable set of complex variables \(\{z_1,z_2,\ldots,z_i,\ldots\}\) is called the conjugate function of the element \(\varphi(x)\), if it is defined in the following way:

\[ \Phi^{\varphi}\{z_1,z_2,\ldots,z_i,\ldots\} = \frac{1}{\pi}\int_{-\pi}^{\pi} M(x;z_1,z_2,\ldots,z_i,\ldots)\overline{\varphi(x)}\,dx. \]

Take for the function \(\varphi(x)\in L_2'[-\pi,\pi]\) the Fourier series \(\varphi(x)=\sum_{k=1}^{\infty} b_k\sin kx\). Then the conjugate function for the element \(\varphi(x)\) will have the form

\[ \Phi^{\varphi}(z_1,z_2,\ldots,z_i,\ldots)=\sum_{k=1}^{\infty} b_k^{*}\mu(k). \tag{8} \]

Thus, to each function \(\varphi(x)\in L_2'[-\pi,\pi]\) one can assign the series

\[ \sum_{k=1}^{\infty} b_k\mu(k). \tag{9} \]

Consider the set of sequences of complex numbers \(\{z_1,z_2,\ldots,z_i,\ldots\}\) satisfying the condition \(|z_i|\le p_i^{1+\varepsilon}\) \((\varepsilon>0)\), where \(p_i\) is the \(i\)-th prime number. We shall denote this set by \(G_1\). Obviously, \(G\subset G_1\).

Definition. Let a function \(\varphi(x)\in L_2'[-\pi,\pi]\) be given,

\[ \varphi(x)=\sum_{k=1}^{\infty} b_k\sin kx. \]

The system \(\{\varphi(nx)\}\) is called close to the trigonometric system \(\{\sin nx\}\), or a \(T\)-system, if the coefficients \(\{b_n\}\) satisfy the conditions:

\[ \text{The series (9) converges absolutely in the domain } G_1; \]

\[ \text{The series (9) has no zeros in the domain } G_1. \tag{10} \]

Remark. Conditions (10) are fulfilled, for example, in the case when

\[ |b_1|>\sum_{k=2}^{\infty}|b_k|\,k^{1+\varepsilon}. \tag{11} \]

The converse is not true, as the following example shows: there exist functions \(\varphi(x)\) for which conditions (10) are fulfilled, while condition (11) is not.

Example 1.

\[ \varphi(x)=\sum_{k=0}^{\infty}\frac{\sin 2^k x}{k!}\quad (0!=1); \qquad \sum_{k=1}^{\infty} b_k\mu(k)=\sum_{k=0}^{\infty}\frac{\mu(2^k)}{k!} = \sum_{k=0}^{\infty}\frac{z^k}{k!}=e^z. \]

It is clear that here conditions (10) are fulfilled, but condition (11) is not fulfilled.

We shall now formulate conditions sufficient for the solution of the functional equation (5).

Lemma 1. In order that the functional equation

\[ F(x)=\sum_{k=1}^{\infty} b_k f(kx), \]

where \(F(x)\) is a given function from \(L_2'[-\pi,\pi]\), \(b_1,b_2,\ldots,b_n,\ldots\) are given numbers, and \(f(x)\) is an unknown function, have a solution, it is sufficient that the series defining the function of a countable set of complex variables

\[ \Phi(z_1,z_2,\ldots,z_i,\ldots)=\sum_{k=1}^{\infty} b_k\mu(k) \]

converge absolutely in the domain \(G_1\{ |z_i|\le p_i^{1+\varepsilon}\}\), and that the function \(\Phi(z_1,z_2,\ldots,z_i,\ldots)\) have no zeros in the same domain. Under these conditions the solution can be represented in the form

\[ f(x)=\sum_{k=1}^{\infty} A_k F(kx), \]

where the coefficients \(A_k\) satisfy the condition

\[ \sum_{k=1}^{\infty}|A_k|\,k^{\varepsilon_1}<+\infty\qquad(\varepsilon_1<\varepsilon). \]

With the aid of Lemma 1 one can obtain the following results.

Theorem 1. Whatever the function \(F(x)\in L'_2[-\pi,\pi]\), the Fourier series of this function with respect to the \(T\)-system \(\{\varphi(nx)\}\) converges in mean to \(F(x)\), and the series formed from the sum of the squares of the Fourier coefficients of this function converges.

Corollary. It follows from Theorem 1 that the \(T\)-system \(\{\varphi(nx)\}\) is a complete system in the space \(L'_2[-\pi,\pi]\) and that, moreover, it is a Riesz basis in this space.

Theorem 2. In order that the Fourier series (2) of a continuous function \(F(x)\) with respect to the \(T\)-system \(\{\varphi(nx)\}\) converge uniformly to \(F(x)\), it is necessary and sufficient that the corresponding trigonometric series (3) converge uniformly.

It follows in particular from Theorem 2 and Lemma 1 that if the function \(F(x)\) satisfies a Lipschitz condition of order \(\alpha\) \((0<\alpha\leqslant 1)\), then the Fourier series of this function with respect to the \(T\)-system converges uniformly to this function.

Theorem 3. In order that the Fourier series (2) of a continuous function \(F(x)\) with respect to the \(T\)-system \(\{\varphi(nx)\}\) converge at every point to \(F(x)\) and that its partial sums be uniformly bounded, it is necessary and sufficient that the corresponding trigonometric series (3) converge at every point to the continuous function \(f(x)\) and that its partial sums be uniformly bounded.

Theorem 4. In order that the Fourier series (2) of a function \(F(x)\in L'_2\) with respect to the \(T\)-system \(\{\varphi(nx)\}\) converge almost everywhere to \(F(x)\) and that its partial sums have an integrable majorant, it is necessary and sufficient that the corresponding trigonometric series (3) converge almost everywhere to \(f(x)\in L'_2[-\pi,\pi]\) and that its partial sums have an integrable majorant.

It follows from Theorem 4 that, for series of the form \(\sum_{k=1}^{\infty} a_k\varphi(kx)\) with respect to the \(T\)-system \(\{\varphi(nx)\}\), the Kolmogorov–Seliverstov–Plessner theorem is valid.

Theorem 5. Let a \(T\)-system \(\{\varphi(nx)\}\) be given. If the series

\[ \sum_{k=2}^{\infty} a_k^2 \ln k \]

converges, then the series (2) converges almost everywhere. In particular, if the series (2) is the Fourier series of a function \(F(x)\in L'_2\), then it converges almost everywhere to \(F(x)\).

A similar result, asserting only almost-everywhere convergence of the series (2) for systems \(\{\varphi(nx)\}\) with the condition

\[ \sum_{k=1}^{\infty} |b_k|<+\infty, \]

which does not ensure completeness of the system \(\{\varphi(nx)\}\), was obtained by V. F. Gaposhkin*.

For summation of the series (2) by the Toeplitz method, theorems analogous to Theorems 2, 3, and 4 are valid. In particular, the following theorem holds:

Theorem 6. Whatever the odd, everywhere continuous, periodic function \(F(x)\) with period \(2\pi\), the Fourier series of this function with respect to the \(T\)-system \(\{\varphi(nx)\}\), by the \((C,1)\) method, is uniformly summable to this function.

Received
10 VII 1957

REFERENCES

¹ A. Wintner, Am. J. Math., 69, 758 (1947).
² P. Hartman, Am. J. Math., 60, 66 (1938).
³ D. G. Bourgin, C. W. Mendel, Trans. Am. Math. Soc., 57, 332 (1945).
⁴ O. Szasz, Trans. Am. Math. Soc., 62, 213 (1947).
⁵ V. Ya. Kozlov, DAN, 61, No. 6 (1948).

* Unpublished.