Frame analysis and approximation in reproducing kernel Hilbert spaces

Abstract

We consider frames F in a given Hilbert space, and we show that every F may be obtained in a constructive way from a reproducing kernel and an orthonormal basis in an ambient Hilbert space. The construction is operator-theoretic, building on a geometric formula for the analysis operator defined from F . Our focus is on the infinite-dimensional case where a priori estimates play a central role, and we extend a number of results which are known so far only in finite dimensions. We further show how this approach may be used both in constructing useful frames in analysis and in applications, and in understanding their geometry and their symmetries.2000 Mathematics Subject Classification. 33C50, 42C15, 46E22, 47B32.

STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

“…Our study of representations of the Cuntz relations [5,6] is closely tied with the study of endomorphisms of B(H), where H is a separable (infinitely dimensional) complex Hilbert space. For treatments of this in the literature, we refer to the paper [8] and the references cited there; and there are connections of our present results to wavelets [1,2,4,7,15,16,18]. For the theory of q-deformations and its varied applications, see, for example, [20,22,23].…”

Markov measures and extended zeta functions

“…To make our paper more accessible, we offer below a few pointers to the relevant literature. Readers familiar with one of these areas, but perhaps not the others, may wish to check the following references covering aspects of these areas used below: operator algebras, Cuntz algebras and their representations ( [13], [4], [10], [8], [16], [31], [22], [24], [25], [26], [40]) ; multiresolutions and their diverse uses ( [1], [12], [3], [23], [2], [14] , [27] ) ; Zeta functions ( [6], [39], [20], [38], [37], [36]); and Markov measures ( [32], [33], [21], [5], [51], [52], [7], [9]). We further use results from harmonic analysis, such as ( [19], [29], [15], [17], [18], [49]).…”

On Fractional Brownian Motion and Wavelets

“…Later papers on this theme include [3][4][5][6] and references therein, to mention only a few. We will have occasion to use the following references regarding C * -algebras [7][8][9][10][11][12][13][14][15][16]; on multiresolutions and wavelets [4,10,[17][18][19][20][21][22][23][24][25][26][27][28][29]; and results from harmonic analysis, [30][31][32][33][34][35][36][37][38]; on Hilbert and reproducing kernels [3,[39][40][41][42]; on Markov measures [1,2,5,6,11,[43][44]…”