Parseval Frames of Some Hilbert Spaces of Entire Vector Valued Functions via Naimark Dilation

Abstract

In this paper, we establish a matrix representation of a reproducing kernel corresponding to a de Branges matrix generated via matrix valued function multiplication, which generalises a well-known result. We utilise this technique to extend certain results on de Branges spaces of scalar valued functions to vector valued functions. By connecting the fundamental construction with Naimark dilation of frames, we discover certain existence and structural characteristics of Parseval frames of normalised reproducing kernels in vector valued de Branges spaces. The dilation arises when certain de Branges spaces of vector valued functions are embedded in a larger de Branges space of vector valued functions. In addition, the embedding translates the kernel functions associated with a frame sequence in the original space into a Riesz basis for the embedding space.