ON THE STRONG REGULARITY OF STATIONARY RANDOM FIELDS

Let f(\lambda)=\left\\beginarrayl 1,-\pi \leqslant \lambda \leqslant 0, \\ \left(\lambda-\frac\pi2^n\right) / \frac\pi2^n+1,, \quad \frac\pi2^n \leqslant \lambda \leqslant \frac3 \pi2^n+1 \\ \left(\frac\pi2^n-1-\lambda\right) / \frac\pi2^n+1, \quad \frac3 \pi2^n+1 \leqslant \lambda \leqslant \frac\pi2^n-1, \quad n=1,2, \endarray\right. There exists a H_2 function A(z)=\sum_n=0^\infty a_n z^n such that \left|A\left(e^i \lambda\right)\right|^2=f(\lambda). Suppose \u(m, n)\_m, n=0, \pm 1, \ldots be a 2-dimensional white noise. Then X(s, t)=\sum_n=0^\infty a_n u(s+n, t-n) is a stationary field. It is strongly singular, i.e. \bigcap \mathscrL\X(m, n), m \leqslant s \text or n \leqslant t\=\mathscrL\X(m, n), m, 3=0, \pm 1, \cdots\ where \mathscrL\\cdots\ is the closed linear space generated by \\cdots\. At the same time its spectral measure is absolutely continous and its spectral deusity is log-integrable. Our example answered an open problem raised in 2.