FRAMES AND BASES

Priyanka, Priyanka and D, Venku Naidu (2018) FRAMES AND BASES. Masters thesis, Indian Institute of Technology, Hyderabad.

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Abstract

The aim of this Project is to present the central parts of the theory of Frames and Bases. A basis B of a vector space V over a �eld F is a linearly independent subset of V that spans V . In fact, a basis provides us with an expansion of all vectors in terms of elementary building blocks and thereby helps us by reducing many questions concerning general vectors to similar questions concerning only the basis elements. However, the conditions to a basis are very restrictive: we require the elements to be linearly independent, and very often even want them to be orthogonal with respect to an inner product. This makes it hard or even impossible to �nd bases satisfying extra conditions, and this is the reason that one might wish to look for a more exible tool. Frames are such tools. A frame for a vector space equipped with an inner product also allows each vector in the space to be written as a linear combination of the elements in the frame, but linear independence between the frame elements is not required. A frame is a generalization of a Hilbert space basis that is not necessarily linearly independent. A frame allows us to represent any vector as a set of frame coe�cients, and to reconstruct a vector from its coe�cients in a numerically stable way. We often aim to decompose a function in terms of functions that are not linearly independent. For example: windowed Fourier transforms, wavelet transforms, non-uniform sampling. Every basis is a frame but any frame may or may not be basis. In the previous semester, I studied the concept of frames in �nite-dimensional vector spaces with inner product spaces. In in�nite-dimensional vector spaces and sequences, I studied Bessel sequences in Hilbert spaces and Riesz bases. In this semester, I worked with explicit constructions in L2-spaces. I studied the concept of Gram matrix and a few applications. Moreover, I worked on frames in Hilbert Spaces in which I studied Riesz bases, characterization of frames, perturbation of frames and a wonderful theorem which says that if ffkg1 k =1 is a frame for H, then several equivalent conditions hold. Finally, I studied frames of translates in which I read about the canonical dual frame. In Gabor frames, I studied Irregular wavelet and basic Gabor frame theory.

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