UDC 513.88

MATHEMATICS

V. P. GURARIIA

ON THE STRUCTURE OF PRIMARY IDEALS IN RINGS OF FUNCTIONS INTEGRABLE WITH AN INCREASING WEIGHT ON THE HALF-LINE

(Presented by Academician V. I. Smirnov on 13 IV 1967)

Denote by \(\mathcal L_\varphi(R)\) the Banach space of functions defined on the real axis \(R\) with norm

\[ \|f\|=\int_{-\infty}^{\infty}|f(t)|\varphi(t)\,dt, \]

where the weight \(\varphi(t)\) satisfies the following conditions:

1. \(\varphi(t)\geqslant 1\).
2. \(\varphi(t+\tau)\leqslant \varphi(t)\varphi(\tau),\quad -\infty<t,\tau<\infty.\)
3. \[ \int_{-\infty}^{\infty}\frac{\ln\varphi(t)}{1+t^2}\,dt<\infty. \]

Let, further, \(\mathcal L_\varphi^\infty(R)\) be the space conjugate to \(\mathcal L_\varphi(R)\). It consists of functions \(g(t)\) for which

\[ \|g\|=\operatorname{ess\,sup}|g(t)|/\varphi(t)<\infty. \]

By virtue of conditions 1—3, after introducing the convolution operation

\[ f_1*f_2=\int_{-\infty}^{\infty} f_1(t-\tau)f_2(\tau)\,d\tau \]

the space \(\mathcal L_\varphi(R)\) becomes a commutative normed regular ring. The space of maximal ideals of this ring is homeomorphic to the real axis, so that every maximal ideal \(M(x_0)\) consists of all functions \(f(t)\in\mathcal L_\varphi(R)\) whose Fourier transform vanishes at the point \(x_0\) \((^1)\).

The question of the structure of the primary ideals of the ring \(\mathcal L_\varphi(R)\) is considerably more delicate. In view of the absence of a unit in \(\mathcal L_\varphi(R)\), one must, generally speaking, consider two types of primary ideals: primary ideals \(I(\infty)\) corresponding to the infinitely distant point of the real axis, i.e. proper closed ideals which are not contained in any maximal ideal \(M(x)\), and primary ideals \(I(x_0)\subset M(x_0)\), i.e. proper closed ideals not contained in any maximal ideal \(M(x)\), \(x\ne x_0\), nor in any primary ideal \(I(\infty)\). As Berling showed \((^2)\), for the rings \(\mathcal L_\varphi(R)\) Wiener’s approximation theorem in its formulation for the case \(\varphi(t)\equiv 1\) is valid, and consequently (see \((^1)\), p. 237), in them there are no primary ideals corresponding to the infinitely distant point.

In studying the primary ideals \(I(x)\) (one may assume, without loss of generality, that \(x=0\)) it is immediately found that the sets \(I_k(0)\) of functions from \(\mathcal L_\varphi(R)\)

\[ I_k(0)=\left\{f:\int_{-\infty}^{\infty} f(t)t^j\,dt=0,\ j=0,\ldots,k\right\} \tag{1} \]

form a chain of primary ideals belonging to the maximal ideal \(M(0)=I_0(0)\), finite or infinite depending on the rate of growth of \(\varphi(t)\) as \(|t|\to\infty\). Moreover, it turns out that in the case of an infinite chain \(\{I_k(0)\}\) the annihilator \(I^\perp\subset \mathscr{L}_\varphi(R)\) of any primary ideal \(I\subset \bigcap I_k(0)\) consists of functions extendable to the whole complex plane as entire functions of zero degree (see, for example, \((^3)\)).*

In the present note we consider the subring \(\mathscr{L}_\varphi(R^+)\) of the ring \(\mathscr{L}_\varphi(R)\), which consists of functions equal to zero on the left half-axis. In the note \((^4)\) all primary ideals of the ring \(\mathscr{L}_\varphi(R^+)\) were described in the case when \(\varphi(t)\equiv 1\). The results given there are readily carried over to the case of the ring \(\mathscr{L}_\varphi(R^+)\), where the weight \(\varphi(t)\) has no more than power growth as \(t\to\infty\). Therefore to conditions 1—3, which the weight \(\varphi(t)\) satisfies, we add the condition

\[ \text{4.}\quad \lim_{t\to\infty}\frac{t^k}{\varphi(t)}=0\quad (k=1,2,\ldots). \]

The Fourier transform of a function \(f(t)\in \mathscr{L}_\varphi(R^+)\)

\[ F(z)=\frac{1}{\sqrt{2\pi}}\int_0^\infty f(t)e^{-itz}\,dt,\qquad \operatorname{Im} z\leqslant 0, \]

is an analytic function in the half-plane \(\operatorname{Im} z<0\) and is infinitely differentiable in the closed half-plane \(\operatorname{Im} z\leqslant 0\).

The maximal ideals \(M(z_0)\) of the ring \(\mathscr{L}_\varphi(R^+)\) may be identified with the points \(z_0\) of the closed lower half-plane; here each maximal ideal \(M(z_0)\) consists of all functions \(f(t)\in \mathscr{L}_\varphi(R^+)\) whose Fourier transform \(F(z)\) vanishes at \(z=z_0\).

As in the case \(\varphi(t)\equiv 1\) \((^4)\), there exist primary ideals of three types: primary ideals \(I(\infty)\), corresponding to the infinitely remote point; primary ideals \(I(z)\), corresponding to a point \(z\) of the open half-plane; and primary ideals \(I(x)\), corresponding to a point \(x\) of the real axis.

In the article \((^5)\) the following approximation theorem was proved in the space \(\mathscr{L}_\varphi(R^+)\), where the weight \(\varphi(t)\) satisfies conditions 1—3.

Let \(\mathfrak{M}\) be a family of functions \(\{f_\alpha(t)\}\) of the space \(\mathscr{L}_\varphi(R^+)\). For the completeness in \(\mathscr{L}_\varphi(R^+)\) of the system of all finite linear combinations of the form
\[ \sum_{\alpha,\beta} C_{\alpha\beta} f_\alpha(t-\tau_{\alpha\beta}), \]
where \(f_\alpha(t)\in \mathfrak{M}\), \(\tau_{\alpha\beta}\geqslant 0\), it is necessary and sufficient that two conditions be fulfilled: 1) there is no interval \((0,\gamma)\), adjoining zero, on which every function in \(\mathfrak{M}\) is equal to zero almost everywhere; 2) the Fourier transforms of all functions in \(\mathfrak{M}\) do not vanish simultaneously at any point of the closed half-plane \(\operatorname{Im} z\leqslant 0\).

This theorem makes it possible to describe completely all primary ideals corresponding to the infinitely remote point and to points of the open half-plane.

For each \(\gamma>0\) introduce the subset \(I_\gamma(\infty)\) of functions of the ring \(\mathscr{L}_\varphi(R^+)\) that vanish almost everywhere on the interval \((0,\gamma)\). It is clear that \(\{I_\gamma(\infty)\}_{\gamma>0}\) is a chain, ordered by inclusion, of primary ideals corresponding to the infinitely remote point. Let \(I(\infty)\) be some primary ideal corresponding to the infinitely remote point, and let \((0,\gamma_I)\), \(\gamma_I\geqslant 0\), be the maximal interval adjoining zero on which all functions in \(I(\infty)\) vanish almost everywhere. The approximation theorem, applied to the family \(\{f(t+\gamma_I)\}\) of all functions \(f(t)\in I(\infty)\), shows that \(I(\infty)=I_\gamma(\infty)\) for \(\gamma=\gamma_I\), i.e., that it is true

* A detailed study of primary ideals in \(L_\varphi(R)\) was carried out in \((^{12})\), though under very stringent additional restrictions on the weight \(\varphi(t)\).

Theorem 1. \(\{I_\gamma(\infty)\}_{\gamma>0}\) is a maximal chain of primary ideals corresponding to the infinitely remote point. Moreover,
\[ \bigcap_\gamma I_\gamma(\infty)=\{0\}. \]

Denote
\[ I_n(z_0)=\{f\in M(z_0)\ (\operatorname{Im} z_0<0):\ F^{(k)}(z_0)=0,\quad k=0,1,\ldots,n\}. \]
\(\{I_n(z_0)\}_{n=1}^{\infty}\) is a chain of primary ideals corresponding to the point \(z_0\), \(\operatorname{Im} z_0<0\). If one uses the fact that for every function \(f(t)\in I_n(z)\), \(F(z)/(z-z_0)^n\) is the Fourier transform of some function from \(\mathscr L_\varphi(R^+)\), then from the approximation theorem one obtains

Theorem 2. \(\{I_n(z_0)\}_1^\infty\) is a maximal chain of primary ideals belonging to the maximal ideal \(M(z_0)\). Moreover,
\[ \bigcap_1^\infty I_n(z_0)=\{0\}. \]

The situation becomes considerably more complicated in the study of primary ideals corresponding to a point \(x\) of the real axis. (Since the mapping \(f\mapsto fe^{-itx}\) is an isomorphic mapping of the primary ideal \(I(x)\subset M(x)\) onto the primary ideal \(I(0)\subset M(0)\), in what follows one may restrict oneself to the study of primary ideals contained in the maximal ideal \(M(0)\).)

In contrast to the case of the whole axis, where a discrete chain of primary ideals of the form (1) was naturally associated with the point \(0\), in the space \(\mathscr L_\varphi(R^+)\) with the point \(0\) there are associated, generally speaking, two chains of primary ideals. One of them, the discrete one, \(\{I_n(0)\}_1^\infty\), consists of the primary ideals
\[ I_n(0)=\left\{f\in M(0):\int_0^\infty f(t)t^k\,dt=0,\quad k=0,1,\ldots,n\right\}. \]

The other, continuous and ordered by inclusion, chain \(\{I_\alpha(0)\}_{\alpha>0}\) consists of the primary ideals
\[ I_\alpha(0)=\left\{f\in M(0):\int_0^\infty f(t)g(t)\,dt=0,\quad \forall g(t)\in B_{1/2,\alpha,\varphi}\right\}. \tag{2} \]

By \(B_{1/2,\alpha,\varphi}\) we denote the subspace of \(\mathscr L_\varphi^\infty(R^+)\) consisting of functions that coincide almost everywhere with functions extendable to the entire complex plane as entire functions of order \(1/2\) and of type less than or equal to \(\alpha\).

Let us consider two primary ideals
\[ \widetilde I(0)=\bigcap_{n>1} I_n(0),\qquad \underset{\sim}{I}(0)=\overline{\bigcup_{\alpha>0} I_\alpha(0)}. \]

In view of condition 4, the space \(\mathscr L_\varphi^\infty(R^+)\) contains all polynomials, and therefore every primary ideal of the first chain is a proper primary ideal containing all primary ideals of the second. Thus,
\[ \widetilde I(0)\supset \underset{\sim}{I}(0). \]
The nontriviality of the second chain essentially depends on the rate of growth of the weight \(\varphi(t)\) as \(t\to\infty\).

Theorem 3. Suppose the weight \(\varphi(t)\) satisfies the condition
\[ \int_1^\infty \frac{\ln\varphi(t)}{t^{3/2}}\,dt<\infty. \tag{3} \]

Then \(\{I_n(0)\}_1^\infty\) and \(\{I_\alpha(0)\}_{\alpha>0}\) are strictly inclusion-ordered maximal chains of primary ideals contained in the maximal ideal \(M(0)\), in the sense that every primary ideal \(I(0)\subset M[0]\) either coincides with one of the ideals of the first or second chain, or
\[ \underset{\sim}{I}(0)\subset I(0)\subset \widetilde I(0). \]

Theorem 4. If the weight \(\varphi(t)\) satisfies the condition

\[ \int_{0}^{\infty}\frac{\ln \varphi(t)}{t^{3/2}}\,dt=\infty, \tag{4} \]

then \(I_{\alpha}(0)=\{0\}\) for every \(\alpha>0\).

Formula (2), which determines the primary ideals of the second chain, makes it possible, from the form of the function \(f(t)\), to judge its membership in the primary ideal \(I_{\alpha}(0)\). Moreover, the nontriviality of \(I_{\alpha}(0)\) under condition (3) is by no means obvious. It turns out that the following criterion holds for membership of a function \(f(t)\) in the primary ideal \(I_{\alpha}(0)\), making it easy to construct nontrivial functions from \(I_{\alpha}(0)\).

Theorem 5. In order that \(f(t)\) belong to the primary ideal \(I_{\alpha}(0)\), it is necessary and sufficient that, for the function

\[ \widetilde f(z)=\frac{1}{2\pi i}\int_{0}^{\infty}\frac{f(t)}{t-z}\,dt, \]

analytic in the complex \(z\)-plane cut along the positive ray of the real axis, the inequality

\[ \varlimsup_{t\to-\infty}\left(\frac{\ln|\widetilde f(t)|}{\sqrt{-t}}\right)\leq -\alpha \]

hold.

The question of the structure of the primary ideals \(I(0)\) for which the strict inclusions

\[ \underline I(0)\subset I(0)\subset \widetilde I(0) \tag{5} \]

hold is connected with the “irregularity” of the weight \(\varphi(t)\). The point is that if the weight \(\varphi(t)\) is “regular,” then ideals satisfying condition (5) do not exist. Namely, the following theorems hold.

Theorem 6*. Let the weight \(\varphi(t)\) satisfy condition (3), and let there exist a sequence of positive polynomials \(P_n(t)\), uniformly bounded in the norm of the space \(\mathcal L_{\varphi}^{\infty}(R^+)\), such that \(P_n(t)\to\varphi(t)\), \(0\leq t<\infty\). Then \(\widetilde I(0)=I(0)\).

Theorem 7. If the logarithmically convex minorant (see \((^{6})\)) of the weight \(\varphi(t)\) satisfies condition (4), then \(\widetilde I(0)=\{0\}\).

In conclusion we note that the questions considered in the article are closely connected with two classical problems of analysis: the quasianalyticity of classes of functions analytic in the half-plane \(\operatorname{Im} z<0\) and infinitely differentiable in the closed half-plane (see, for example, \((^{7})\)), and weighted approximation by polynomials on the half-axis \((^{8,9})\).

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
23 III 1967

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* It is interesting to compare this result with the theorems of the articles \((^{10,11})\), which contain criteria for the completeness of polynomials in the space of entire functions of zero degree.