Dyadic distributions

B. I. Golubov

Sbornik. Mathematics, Tome 198 (2007) no. 2, pp. 207-230

Résumé

On the basis of the concept of pointwise dyadic derivative dyadic distributions are introduced as continuous linear functionals on the linear space $D_d(\mathbb R_+)$ of infinitely differentiable functions compactly supported by the positive half-axis $\mathbb R_+$ together with all dyadic derivatives. The completeness of the space $D'_d(\mathbb R_+)$ of dyadic distributions is established. It is shown that a locally integrable function on $\mathbb R_+$ generates a dyadic distribution. In addition, the space $S_d(\mathbb R_+)$ of infinitely dyadically differentiable functions on $\mathbb R_+$ rapidly decreasing in the neighbourhood of $+\infty$ is defined. The space $S'_d(\mathbb R_+)$ of dyadic distributions of slow growth is introduced as the space of continuous linear functionals on $S_d(\mathbb R_+)$. The completeness of the space $S'_d(\mathbb R_+)$ is established; it is proved that each integrable function on $\mathbb R_+$ with polynomial growth at $+\infty$ generates a dyadic distribution of slow growth. Bibliography: 25 titles.