V. P. MIKHAILOV

ON RIESZ BASES IN \(\mathscr{L}_2(0,1)\)

(Presented by Academician I. G. Petrovskii on 23 I 1962)

A sequence \(\varphi_1(x), \ldots, \varphi_n(x), \ldots,\ x \in (0,1)\), from \(\mathscr{L}_2(0,1)\) is called a basis in \(\mathscr{L}_2(0,1)\) if for every element \(f(x)\in \mathscr{L}_2(0,1)\) there exists a unique expansion

\[ f=c_1\varphi_1+\cdots+c_n\varphi_n+\cdots, \tag{1} \]

converging to \(f\) in the mean.

Following N. K. Bari \((^1)\), we shall call the basis \(\varphi_1,\ldots,\varphi_n,\ldots\) a Riesz basis if there exists a constant \(\gamma \ge 1\) such that, for every \(f(x)\in \mathscr{L}_2(0,1)\), the inequalities

\[ \gamma^{-1}\sum |c_k|^2 \le \|f\|^2 \le \gamma \sum |c_k|^2 \tag{2} \]

hold.

Consider the ordinary linear differential operator of order \(n\)

\[ l(y)=y^{(n)}+p_1(x)y^{(n-1)}+\cdots+p_n(x)y, \tag{3} \]

defined on the interval \([0,1]\). We shall assume that the coefficients \(p_s(x)\) are summable on \((0,1)\) together with their derivatives up to order \((n-s)\), \(s=1,\ldots,n\). We are interested in the problem of eigenfunctions of the operator \(l(y)\) under the boundary conditions

\[ U_\nu(y)\equiv U_{\nu0}(y)+U_{\nu1}(y)=0,\qquad \nu=1,\ldots,n, \tag{4} \]

\[ U_{\nu0}(y)=\alpha_\nu y_0^{(k_\nu)}+\alpha_{\nu,k_\nu-1}y_0^{(k_\nu-1)}+\cdots;\quad U_{\nu1}(y)=\beta_\nu y_1^{(k_\nu)}+\beta_{\nu,k_\nu-1}y_1^{(k_\nu-1)}+\cdots; \]

\[ |\alpha_\nu|+|\beta_\nu|\ne0;\qquad n-1\ge k_1\ge k_2\ge\cdots\ge k_n;\qquad k_{\nu+2}<k_\nu;\qquad y_0^{(s)}=\left.\frac{d^s y(x)}{dx^s}\right|_{x=0},\quad y_1^{(s)}=\left.\frac{d^s y(x)}{dx^s}\right|_{x=1}. \]

The system adjoint to \(\{\varphi_k\}\) is the system of eigenfunctions \(\{\varphi_k^*\}\) of the operator \(l^*(y)\) (adjoint to \(l(y)\)) under the boundary conditions \((4^*)\), adjoint to the conditions (4). (We note that \(\varphi_k^*\) corresponds to the eigenvalue \(\lambda_k^*=\overline{\lambda}_k\), where \(\lambda_k\) is the eigenvalue corresponding to \(\varphi_k\).) In what follows, the functions \(\varphi_k\) and \(\varphi_k^*\) will be assumed normalized so that \((\varphi_k,\varphi_j^*)=\delta_{kj}\), \(k,j=1,2,\ldots\)

The conditions (4) are called regular \((^{2-4})\) if certain determinants \(\theta_0\) and \(\theta_1\) for odd \(n\), and \(\theta_{-1}\), \(\theta_1\) for even \(n\), are nonzero \(((^4), p. 51)\). We note that \(\theta_0,\theta_1,\theta_{-1}\), and hence also the regularity of the conditions (4), depend only on the coefficients \(\alpha_\nu\) and \(\beta_\nu\), \(\nu=1,\ldots,n\), at the highest derivatives in the conditions (4).

We shall say that the conditions (4) are strongly regular if they are regular and, in addition, for even \(n\), \(\theta_0^2\ne4\theta_1\theta_{-1}\).

In \((^3)\) it was in fact established that the root functions \(\{\varphi_k\}\) of the operator \(l(y)\) under strongly regular boundary conditions (4) form a basis in \(\mathscr{L}_2(0,1)\).

The purpose of the present note is to prove the following assertion:

Theorem. The totality of root functions of the operator (3) under stren—

regular boundary conditions (4), forms a Riesz basis in the space \(\mathscr L_2(0,1)\).

Since, under the hypotheses of the theorem, the eigenvalues (except for a finite number of them) are simple, it is clearly sufficient to carry out the proof of the theorem for the case when all eigenvalues are simple and, thus, the root functions coincide with the eigenfunctions.

In \((^{5,6,10})\) more general problems were considered for certain operators. As applied to our problem, the results of these papers, as well as of \((^{11})\), are contained in the theorem formulated above as special cases.

Let us note that the theorem remains valid also in the case when the coefficients \(l(y)\) in (3) are, as in Birkhoff \((^{2})\), analytic functions of \(\lambda\), and when, as in Tamarkin \((^{3})\), certain integral operators are added to the boundary conditions (4).

The proof of the theorem is based on several lemmas.

Lemma 1. If the conditions (4) for the operator \(l(y)\) are regular (strongly regular), then the adjoint conditions \((4^*)\) for the adjoint operator \(l^*(y)\) are also regular (strongly regular).

By a direct verification we see that the principal coefficients in the conditions \((4^*)\) are determined only by the principal coefficients of (4). Therefore the proof of Lemma 1 can be carried out by comparing the asymptotic formulas for the eigenvalues \(\lambda_k\) and \(\lambda_k^*\) as \(|k|\to\infty\) \((^{4})\).

Consider, for \(f(\xi)\in \mathscr L_2(0,\infty)\), the Hilbert transform \((^{7})\)

\[ F(x)=\int_0^\infty \frac{f(\xi)\,d\xi}{\xi-x}, \tag{5} \]

which is an analytic function of \(x\) for \(\operatorname{Im}x>0\) and for \(\operatorname{Im}x<0\). Denote by \(\|F\|_{\mathscr L_2(0,\infty e^{i\psi})}^2\) the integral

\[ \int_0^\infty |F(\rho e^{i\psi})|^2\,d\rho . \]

Lemma 2. For every \(\psi\), \(F(x)\in \mathscr L_2(0,\infty e^{i\psi})\), and for \(\psi\equiv0\pmod\pi\)

\[ \|F\|_{\mathscr L_2(0,\infty e^{i\psi})} = \|f\|_{\mathscr L_2(0,\infty)}, \]

while for \(\psi\not\equiv0\pmod\pi\)

\[ \|F\|_{\mathscr L_2(0,\infty e^{i\psi})} \le \sqrt{\frac{\pi}{|\sin\psi|}}\, \|f\|_{\mathscr L_2(0,\infty)} . \]

If \(\psi\equiv0\pmod\pi\), then Lemma 2 is known \((^{1})\). For what follows it is convenient to introduce into consideration the kernel

\[ K_N(x,\xi)=\sum_{k=0}^N e^{-(\alpha+i\beta)xk-(\gamma+i\delta)\xi k}, \tag{6} \]

where \(N\) is some natural number; \(\alpha,\beta,\gamma,\delta\) are real numbers, \(\alpha\ge0,\ \gamma\ge0\), and \(\beta\) and \(\delta\) are such that, when \(\alpha=0\), \(\beta=2\pi\) or \(\beta=-2\pi\), and when \(\gamma=0\), \(\delta=2\pi\) or \(\delta=-2\pi\), \(x\in[0,1]\), \(\xi\in[0,1]\).

Let \(f(x)\in\mathscr L_2(0,1)\). Then

\[ F_N(x)=\int_0^1 K_N(x,\xi)f(\xi)\,d\xi \]

is an analytic function of \(x\) for every \(N\).

Lemma 3. There exists a constant \(C_0\), independent of \(N\) and of \(f(\xi)\), such that

\[ \|F_N\|\le C_0\|f\|. \tag{7} \]

If \(\alpha=\gamma=0\), and \(\beta=\pm2\pi,\ \delta=\pm2\pi\), then

\[ F_N(x)=c_0+c_1e^{2\pi ix}+\cdots+c_Ne^{2\pi iNx}, \]

where \(c_k=(f,e^{\mp2\pi ik\xi})\). Inequality (7) in this case is...

is a simple consequence of Bessel’s inequality for trigonometric series. Let \(\alpha>0,\ \gamma>0\). Represent \(F_N(x)\) in the form

\[ F_N(x)=\int_0^1 \frac{f(\xi)\,d\xi}{1-e^{-\theta(x,\xi)}}-\int_0^1 \frac{f(\xi)e^{-(N+1)\theta(x,\xi)}}{1-e^{-\theta(x,\xi)}}\,d\xi =F^{(1)}(x)-F_N^{(2)}(x); \tag{8} \]

\[ F^{(1)}(x)=\int_0^\varepsilon \frac{f(\xi)\,d\xi}{1-e^{-\theta(x,\xi)}}+\int_\varepsilon^1 \frac{f(\xi)\,d\xi}{1-e^{-\theta(x,\xi)}} =\Phi_{1\varepsilon}(x)+\Phi_{2\varepsilon}(x), \tag{9} \]

where

\[ \theta(x,\xi)=-(\alpha+i\beta)x-(\gamma+i\delta)\xi =-(\gamma+i\delta)(\xi-Axe^{i\psi}),\quad A=\left|\frac{\alpha+i\beta}{\gamma+i\delta}\right|, \]

\(\psi=\arg(\alpha+i\beta)-\arg(\gamma+i\delta)+\pi\), and \(\varepsilon\) is for the time being an arbitrary number from \((0,1)\). We note that \(\theta(x,\xi)=0\) only when \(x=\xi=0\) (since \(\alpha>0,\ \gamma>0\)). It is verified directly that

\[ \|\Phi_{2\varepsilon}\|\le \|f\|/\sqrt{1-e^{-\varepsilon\gamma}},\qquad \|\Phi_{1\varepsilon}\|_{\mathscr L_2(\varepsilon,1)} \le \|f\|/\sqrt{1-e^{-\varepsilon\alpha}}. \tag{10} \]

Now take \(\varepsilon>0\) so small that for \(x\le\varepsilon,\ \xi\le\varepsilon\) the inequality

\[ |1-e^{-\theta(x,\xi)}-\theta(x,\xi)|\le |\theta^2(x,\xi)| \]

holds. Then, according to (10) and Lemma 2,

\[ \|\Phi_{1\varepsilon}\|^2 =\|\Phi_{1\varepsilon}\|^2_{L_2(0,\varepsilon)} +\|\Phi_{1\varepsilon}\|^2_{L_2(\varepsilon,1)} \le \left(1+\frac{1}{1-e^{-\varepsilon\alpha}} +\frac{\pi}{(\gamma^2+\delta^2)\sin\psi}\right)\|f\|^2 \tag{11} \]

(when applying Lemma 2 we extended the function \(f(\xi)\) from the interval \((0,\varepsilon)\) to the interval \((0,\infty)\) by zero). Taking (10) and (11) into account, we obtain from (9) estimate (7) for \(F^{(1)}(x)\). The estimate for \(F_N^{(2)}(x)\), uniform in \(N\), is obtained analogously.

Let now \(\alpha>0,\ \gamma=0\) (similarly, \(\alpha=0,\ \gamma>0\)); then \(\theta(x,\xi)\) in (9) vanishes at two points of the square \(0\le x\le 1,\ 0\le \xi\le 1\): at the point \(x=\xi=0\) and at the point \(x=0,\ \xi=1\) (similarly at the points \(x=\xi=0,\ x=1,\ \xi=0\)). To obtain estimate (7) in this case, we first divide the interval \(0\le \xi\le 1\) into two parts \((0,1/2)\) and \((1/2,1)\). In each of these parts the required estimate (7) is obtained by the method described above for the case \(\alpha>0,\ \gamma>0\).

Let

\[ \widetilde K_N(x,\xi)=\sum_{k=1}^N \frac1k e^{-(\alpha+i\beta)xk-(\gamma+i\delta)\xi k}, \tag{12} \]

where \(\alpha,\beta,\gamma,\delta,N\) are the same constants as in the kernel (6).

Lemma 4. Let \(f(\xi)\in \mathscr L_2(0,1)\). Then

\[ \widetilde F_N(x)=\int_0^1 \widetilde K_N(x,\xi)f(\xi)\,d\xi \]

belongs to \(\mathscr L_2(0,1)\) for every \(N\), and the estimate

\[ \|\widetilde F_N\|\le C_1\|f\| \tag{13} \]

holds, where the constant \(C_1\) does not depend on \(N\) and \(f(x)\).

Consider now the kernel

\[ R_N(x,\xi)=\sum_{|k|\le N}\varphi_k^*(x)\varphi_k^*(\xi), \tag{14} \]

consisting of eigenfunctions of the operator \(l^*(y)\) under the conditions \((4^*)\).

Lemma 5. Let \(f(x)\in \mathscr L_2(0,1)\). Then

\[ g_N(x)=\int_0^1 R_N(x,\xi)f(\xi)\,d\xi \]

for every \(N\) belongs to \(\mathscr L_2(0,1)\), and the estimate

\[ \|g_N\|\le C_2\|f\| \]

holds, where the constant \(C_2\) does not depend on \(N\) and \(f(\xi)\).

Assume, for definiteness, that the order of the operator (3) is odd, \(n=2\mu-1\). For the normalized eigenfunctions \(\varphi_k^*(x)\) of the adjoint problem, by virtue of Lemma 1 one can write, uniformly with respect to \(x\in[0,1]\), the asympto-

asymptotic in \(k\) as \(|k|\to\infty\) expressions

\[ \varphi_k^*(x)= \sum_{|s|<\nu-1}\left(A_s^{\pm}+\frac{B_s^{\pm}}{k}\right)e^{\omega_s(2k+\sigma)x} \sum_{\substack{|s|\geq \nu\\ s\ne \nu}} \left(A_s^{\pm}+\frac{B_s^{\pm}}{k}\right)e^{\omega_s(2\pi k+\sigma)(1-x)} + \]

\[ +\left(A_\nu^{\pm}+\frac{B_\nu^{\pm}}{k}\right)e^{(2k\pi i+\sigma)x} +O\left(\frac1{k^2}\right), \tag{15} \]

where

\[ \omega_s=\exp\left[i\pi\left(1+\frac1{2n}-\frac{2s}{n}\right)\right],\qquad s=0,\pm1,\ldots; \]

\(\sigma\) is a complex number; \(A_s^+\) and \(A_s^-\) \((B_s^+\) and \(B_s^-)\) are certain constant numbers which should be substituted in (15) in place of \(A_s^{\pm}\) \((B_s^{\pm})\) for \(k>0\) and \(k<0\), respectively; \(B_\nu^\pm\ne0\). By direct verification we see that the kernel \(R_N(x,\xi)\) is represented in the form

\[ R_N(x,\xi)=\sum_{s=1}^{M}\bigl[ D_sK_N^{(s)}(x,\xi)+E_sK_N^{(s)}(1-x,\xi)+G_sK_N^{(s)}(x,1-\xi)+ \]

\[ +H_sK_N^{(s)}(1-x,1-\xi)+\widetilde D_s\widetilde K_N^{(s)}(x,\xi)+\widetilde E_s\widetilde K_N^{(s)}(1-x,\xi)+ \]

\[ +\widetilde G_s\widetilde K_N^{(s)}(x,1-\xi)+\widetilde H_s\widetilde K_N^{(s)}(1-x,1-\xi)\bigr] +\widetilde{\widetilde K}_N(x,\xi), \tag{16} \]

where \(M\) is some natural number; \(D_s,\ldots,\widetilde H_s\) are certain complex numbers, while the functions \(K_N^{(s)}(x,\xi)\) and \(\widetilde K_N^{(s)}(x,\xi)\) are kernels of the type (6) and (12), respectively; the kernel \(\widetilde{\widetilde K}_N(x,\xi)\) is uniformly bounded with respect to \(N\) and uniformly continuous in \(x,\xi\).

Lemma 5 now follows immediately from Lemmas 3 and 4 with the aid of representation (16).

To prove the theorem it remains for us to use a theorem of N. K. Bari \((^1)\).

Bari’s theorem. In order that the basis \(\{\varphi_k\}\) be a Riesz basis, it is necessary and sufficient that there exist a bounded, invertible, Hermitian, positive operator \(A\) taking the system \(\{\varphi_k\}\) into its conjugate system \(\{\varphi_k^*\}\).

Thus, suppose that the operator \(A\) is given on the basis by: \(A\varphi_k=\varphi_k^*\), \(k=1,2,\ldots\). By this it is defined for any

\[ f_N=c_{-N}\varphi_{-N}+\cdots+c_N\varphi_N,\qquad N=0,1,\ldots, \]

\[ c_k=(f,\varphi_k^*),\qquad Af_N=\sum_{k=-N}^{+N}c_k\varphi_k^* =\int_0^1 R_N(x,\xi)f(\xi)\,d\xi, \]

where \(R_N(x,\xi)\) is the kernel (14). By virtue of Lemma 5, the operator \(A\) extends to all of \(\mathscr L_2(0,1)\), and \(\|A\|\leq C_2\). Let \(f\) and \(g\in\mathscr L_2(0,1)\),

\[ f=c_0\varphi_0+\cdots+c_N\varphi_N+\cdots,\qquad g=b_0\varphi_0+\cdots+b_N\varphi_N+\cdots, \]

then

\[ (Af,g)=c_0\bar b_0+\cdots+c_N\bar b_N+\cdots=(f,Ag), \]

i.e. \(A\) is a Hermitian operator.

Moreover,

\[ (Af,f)=|c_0|^2+\cdots+|c_N|^2+\cdots>0, \]

and \((Af,f)=0\) only under the condition \(f=0\). Thus the operator \(A\) satisfies all the conditions of Bari’s theorem. Hence the basis \(\{\psi_k\}\) is a Riesz basis.

Moscow State University
named after M. V. Lomonosov

Received
18 I 1962

CITED LITERATURE

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\(^{2}\) G. D. Birkhoff, Trans. Am. Math. Soc., 9, 373 (1908).
\(^{3}\) Ya. D. Tamarkin, On Some General Problems in the Theory of Ordinary Equations, Petrograd, 1917.
\(^{4}\) M. A. Naimark, Linear Differential Operators, Moscow, 1954.
\(^{5}\) B. R. Mukminov, DAN, 99, No. 4 (1954).
\(^{6}\) I. M. Glazman, UMN, 13, issue 3 (1958).
\(^{7}\) E. T. Titmarsh, Introduction to the Theory of Fourier Integrals, Moscow, 1948.
\(^{8}\) G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Moscow, 1948.
\(^{9}\) I. M. Gelfand, Uch. zap. Mosk. univ., vol. 148, Matemat., 4 (1951).
\(^{10}\) A. S. Markus, DAN, 132, No. 3 (1961).
\(^{11}\) S. Kramer, Pacific J. Math., 7, No. 3, 218 (1957).