演講公告 91年4月15日 (星期一)

摘要

Well before orthonormal wavelet bases existed, non-orthogonal wavelets had been used. Unlike the case of orthonormal wavelets, non-orthogonal wavelet cannot be used in any function space other than $L^2$. In fact, Tchamitchian proved: for every $p>2$, there exists a function $thta(x)$, belonging to the Schwartz claa and all of whose moments are zero. However, it satisfies two apparently contradictory properties as follows: (a) the collection of functions$2^{j/2}\thta(2^jx-k), j, k\in Z$ is a Riesz basis of $L^2$; (b) the above collection is not complete in $L^p$. There is another natural question in non-orthogonal wavelet theory. Daubechies provided the necessary condition on a function which can be used to generate a non-orthogonal wavelet by the dilation and translation. Is this condition also sufficient? In this talk, we will explain three generations of Calderon-Zygmund singular integral operators. In particula, we will show a new result of Calderon-Zygmund operator theory. Using this new result, we will give a partila answer about these two questions mentioned above. Roughly speaking, under a little bit smoothness condition, the Daubechias' condition is sufficient. Moreover, non-orthogonal wavelet provided by these condiitons is complete in $L^p$ for all $1