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}})
{(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).