MATHEMATICS

V. P. GURARII

A GENERALIZATION OF THE FOURIER TRANSFORM AND OF THE WIENER—PALEY THEOREM

(Presented by Academician S. N. Bernstein on 21 X 1959)

§ 1. In the present paper we consider the question of extending the Fourier transform to the Hilbert space \(L_\varphi^2\) of functions, the scalar product in which is equal to

\[ (f,\ g)=\int_{-\infty}^{\infty}\frac{f(x)\overline{g(x)}}{\varphi(x)}\,dx, \]

where the weight \(\varphi(x)\) is a positive function on the real axis. The Fourier transform in such a space was first considered by N. I. Akhiezer \((^{1})\) under the assumption that \(\varphi(z)\) is an entire function of zero order whose zeros lie in the strip \(|\operatorname{Re} z|<A\). In that work the functions

\[ \omega(z)=\prod\left(1-\frac{z}{z_k}\right),\qquad \overline{\omega}(z)=\prod\left(1-\frac{z}{\overline{z_k}}\right), \]

were constructed, where \(z_k\) are the zeros of \(\varphi(z)\) lying in the half-plane \(\operatorname{Im} z>0\), and the following theorems were proved:

I. Every function \(f(x)\in L_\varphi^2\) can be represented in the form

\[ f(x)=\sum_{k=0}^{\infty}a_kP_k(x)+ \frac{\omega(x)}{\sqrt{2\pi}}\int_{0}^{\infty}h(t)e^{itx}dt+ \frac{\overline{\omega}(x)}{\sqrt{2\pi}}\int_{-\infty}^{0}h(t)e^{itx}dt, \]

where \(P_k(x)\) is an orthonormal sequence of polynomials in \(L_\varphi^2\), \(a_k=(f,P_k)\) \((k=0,1,\ldots)\), and \(h(t)\in L^2(-\infty,\infty)\). Moreover, Parseval’s equality holds:

\[ \|f\|^2=\sum_{k=0}^{\infty}|a_k|^2+\|h\|_{L^2}^{\,2}. \]

II. In order that the function \(f(x)\in L_\varphi^2\) be an entire function of finite degree, less than or equal to \(\sigma\), it is necessary and sufficient that \(h(t)\) be equal to zero for \(|t|>\sigma\).

We have proved analogous theorems, but under weaker restrictions on the function \(\varphi(x)\). In addition, it is shown that any further weakening of the requirements on \(\varphi(x)\) is impossible.

§ 2. We shall assume that \(\varphi(z)\) is an entire function of zero order belonging to class \(*A\). Let us note that under these restrictions on the weight, polynomials may fail to belong to \(L_\varphi^2\).

\[ \text{* An entire function is called a function of class } A \text{ if its zeros satisfy the inequality } \sum_k\left|\operatorname{Im}\frac{1}{a_k}\right|<\infty. \]

In what follows we shall rely on the following theorem of N. I. Akhiezer \((^{2})\):

In order that an entire function \(F(z)\) of finite degree \(\sigma\) be representable in the form \(F(x)=|\Omega(x)|^2\), where \(\Omega(z)\) is an entire function of finite degree \(\sigma/2\) with roots in the half-plane \(\operatorname{Im} z \geqslant 0\) \((\operatorname{Im} z \leqslant 0)\), it is necessary and sufficient that \(F(z)\) be a function of class \(A\), nonnegative on the real axis.

Applying this theorem to the weight \(\varphi(x)\), we obtain that \(\varphi(x)=|\omega(x)|^2\), where \(\omega(z)\) is an entire function of zero degree with roots in the half-plane \(\operatorname{Im} z>0\). Let \(\{z_k\}_1^\infty\) be the sequence of roots of \(\omega(z)\). For simplicity of exposition we shall assume that the roots \(z_k\) are simple. Put \(\overline{\omega}(z)=\overline{\omega(\overline{z})}\). The roots of \(\overline{\omega}(z)\) lie in the half-plane \(\operatorname{Im} z<0\). Introduce the functions

\[ \omega_k(z)=\sqrt{\frac{z_k-\overline{z}_k}{2\pi i}\, \frac{\omega(z)(z-\overline{z}_1)(z-\overline{z}_2)\cdots(z-\overline{z}_{k-1})} {(z-z_1)\cdots(z-z_{k-1})(z-z_k)}}\qquad (k=1,2,\ldots); \]

these are entire functions of zero degree belonging to \(L_\varphi^2\). It is easy to verify that \(\{\omega_k(x)\}_1^\infty\) is an orthonormal sequence in \(L_\varphi^2\).

Theorem 1. In order that a function \(f(x)\) belong to the space \(L_\varphi^2\), it is necessary and sufficient that it be representable in the form

\[ f(x)=\sum_{k=1}^{\infty} a_k\omega_k(x) +\frac{\omega(x)}{\sqrt{2\pi}}\int_0^\infty h(t)e^{itx}\,dt +\frac{\overline{\omega}(x)}{\sqrt{2\pi}}\int_{-\infty}^0 h(t)e^{itx}\,dt, \tag{1} \]

where \(a_k=(f,\omega_k)\), \(h(t)\in L^2(-\infty,\infty)\). In this case Parseval’s equality holds

\[ \|f\|^2=\sum_{k=1}^{\infty}|a_k|^2+\|h\|_{L^2}^2 . \tag{2} \]

Proof. We shall confine ourselves to proving necessity, since sufficiency is obvious.

Consider three families of functions:

\[ 1.\ \omega_k(x)\ (k=1,2,\ldots).\qquad 2.\ \frac{\omega(x)}{x-z}\ (\operatorname{Im}z<0).\qquad 3.\ \frac{\overline{\omega}(x)}{x-w}\ (\operatorname{Im}w>0). \]

All these functions belong to the space \(L_\varphi^2\), and the functions from the different systems are mutually orthogonal. Denote by \(H^0, H^+, H^-\), respectively, the closures in \(L_\varphi^2\) of the linear spans of the families 1, 2, 3. These closures form mutually orthogonal subspaces in \(L_\varphi^2\). Note that any function \(f^+(x)\) belonging to \(H^+\) has the form

\[ f^+(x)=\frac{\omega(x)}{\sqrt{2\pi}}\int_0^\infty h(t)e^{itx}\,dt. \tag{3} \]

This becomes clear if one takes into account that the closure of the set of linear aggregates of the form \(\sum_k \dfrac{c_k\omega_k(x)}{x-a_k}\) \((\operatorname{Im}a_k<0)\) in \(L_\varphi^2\) is equivalent to the closure in \(L^2(-\infty,\infty)\) of the aggregates \(\sum_k \dfrac{c_k}{x-a_k}\). Similarly, any function \(f(x)\in H^-\) has the form

\[ f^-(x)=\frac{\overline{\omega}(x)}{\sqrt{2\pi}}\int_{-\infty}^{0} h(t)e^{itx}\,dt. \tag{3'} \]

We shall prove that the direct sum of the subspaces \(H^0\), \(H^+\), and \(H^-\) gives the whole space \(L_\varphi^2\), i.e., that \(H^0 \oplus H^+ \oplus H^- = L_\varphi^2\). To this end it is necessary to show that every function \(f(x)\in L_\varphi^2\) orthogonal to each of the subspaces \(H^+\), \(H^-\), and \(H^0\) is identically zero.

1. Let \(f(x)\in L_\varphi^2\), and let \(f(x)\) be orthogonal to any function from \(H^+\); then, denoting by \(\bar L^2\) (respectively, \(\overset{+}{L}{}^2\)) the space of functions from \(L^2(-\infty,\infty)\) whose Fourier transform is equal to zero for \(t>0\) (respectively \(t<0\)), one may assert that

\[ \frac{f(x)}{\omega(x)}\in \bar L^2 . \tag{4} \]

It is not difficult to prove the converse as well, namely: if (4) holds, then \(f(x)\) is orthogonal to any function from \(H^+\).

1. Analogously we obtain that the condition

\[ \frac{f(x)}{\omega(x)}\in \overset{+}{L}{}^2 \]

is necessary and sufficient for the function \(f(x)\) to be orthogonal to any function from \(H^-\).

Using the well-known Wiener–Paley theorem and results 1, 2, one can show that a function orthogonal to \(H^+\) and \(H^-\) extends to the whole plane as an entire function of zero degree.

1. It remains to prove that an entire function \(f(z)\) of zero degree, for which \(f(x)\in L_\varphi^2\) and \((f,\omega_k)=0\) \((k=1,2,\ldots)\), is identically zero. But from the equalities \((f,\omega_k)=0\) \((k=1,2,\ldots)\), by Cauchy’s theorem it follows that \(f(\bar z_k)=0\). Therefore the function \(\psi(z)=f(z)/\omega(z)\) is an entire function, and its degree is obviously equal to zero.

And since \(\psi(x)\in L^2(-\infty,\infty)\), it follows that \(\psi(z)\equiv 0\), and hence \(f(z)\equiv 0\). Thus, we have proved that every function \(f(x)\in L_\varphi^2\) is represented in the form

\[ f(x)=f^0(x)+f^+(x)+f^-(x), \tag{5} \]

where

\[ f^0(x)\in H^0,\quad f^+(x)\in H^+,\quad f^-(x)\in H^- \quad\text{and}\quad \|f\|^2=\|f^0\|^2+\|f^+\|^2+\|f^-\|^2 . \]

Moreover, we have proved that \(\{\omega_k(x)\}_1^\infty\) is not only an orthonormal system, but also a basis in \(H^0\). From (3), (3′), (5) the assertion of the theorem follows.

It should be noted that the series \(\sum_k a_k\omega_k(z)\), under the condition \(\sum_k |a_k|^2<\infty\), converges uniformly in every finite domain of the complex plane.

§ 3. Theorem 2. In order that \(f(x)\in L_\varphi^2\) be an entire function of finite degree \(\sigma\), it is necessary and sufficient that in the expansion (1) \(h(t)=0\) for \(|t|>\sigma\).

Proof. Here too we shall prove necessity, since sufficiency is obvious.

Preserving the notation for each of the parts of the expansion (5), it is easy to see that \(f^+(x)\) is also an entire function of finite degree, not exceeding \(\sigma\). From representation (3) it is clear that

\[ \int_0^\infty h(t)e^{itx}\,dt \]

is an entire function whose degree also does not exceed \(\sigma\). By the Wiener–Paley theorem \(h(t)=0\) for \(t>\sigma\).

In exactly the same way we obtain that \(h(t)=0\) for \(t<-\sigma\).

§ 4. Theorem 3. Let the weight \(\varphi(x)\) satisfy the following conditions:

1) There exists a function \(\omega(z)\), analytic and having no zeros in the lower half-plane, such that \(\varphi(x)=|\omega(x)|^2\).

2) Every function \(f(x)\in L_\varphi^2\) has the representation
\[ f(x)=f_0(x)+\frac{\omega(x)}{\sqrt{2\pi}}\int_0^\infty h(t)e^{itx}\,dt +\frac{\overline{\omega}(x)}{\sqrt{2\pi}}\int_{-\infty}^0 h(t)e^{itx}\,dt, \]
where \(f_0(x)\) is an entire function of zero degree, \(f_0(x)\in L_\varphi^2\), \(h(t)\in L^2(-\infty,\infty)\), \(\overline{\omega}(z)=\omega(\overline z)\); moreover \(\|f\|^2=\|f_0\|^2+\|h\|_{L^2}^2\).

Then \(\varphi(x)\) extends to the entire complex plane as an entire function of zero degree and of class \(A\).

§ 5. Suppose that all polynomials belong to the space \(L_\varphi^2\). In this case one may pose the question of the completeness of the system of polynomials in the space \(H^0\) of entire functions of zero degree belonging to \(L_\varphi^2\).

Theorem 4. In order that the system of polynomials be complete in \(H^0\), it is necessary and sufficient that, for all \(z\) and \(w\), the equality
\[ \sum_{n=1}^{\infty} P_n(z)\overline{P_n(w)} = \frac{1}{2\pi i}\, \frac{\omega(z)\overline{\omega}(w)-\overline{\omega}(z)\omega(w)}{z-w}, \]
hold, where \(\{P_n(x)\}_1^\infty\) is an orthonormal sequence of polynomials in \(L_\varphi^2\).

Received
20 X 1959

REFERENCES

¹ N. I. Akhiezer, DAN, 96, No. 5, 889 (1954). ² N. I. Akhiezer, DAN, 63, No. 5, 475 (1948).