orthonormal basis

This paper is devoted to introduce a new concept of bases for the range of the operator $K$. Actually, we consider controlled $K$-orthonormal and controlled $K$-Riesz bases which are a generalization of ordinary bases in Hilbert spaces. In the sequel, we give some characterizations of these  sequences. Next, we construct some new controlled bases with operator theory tools.  Finally, some dual characterizations are given.

Let $K$ be a bounded operator. $K$-frames are ordinary frames for the range $K$. These frames are a generalization of ordinary frames and are certainly different from these frames. This research introduces a new concept of bases for the range $K$. Here we define the  $K$-orthonormal basis and the $K$-Riesz basis, and then we describe their properties. As might be expected, the $K$-bases differ from the ordinary ones mentioned in this article.

The main aim of this paper is to construct an orthonormal wavelet on H2(D), the Hardy space of analytic functions on the open unit disc with square summable Taylor coefficients