We make a deep study of the distance between frames and between subspaces of a Hilbert space. There are many surprises here. First, of the six standard ways of measuring distance between subspaces, 5 of them are equal and the sixth, chordal distance, is within a factor of 2 of the others. We also show that the vectors giving chordal distance are biorthogonal which the definition does not indica...

We investigate the frame properties and closedness for the shift invariant space

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With r(d, n) := ( n+d−1 n ) , the first method generates, for each m ≥ n, two families of tight frames in R with r(d + 1,m) elements. The secon...

In this paper, we show that in each nite dimensional Hilbert space, a frame of subspaces is an ultra Bessel sequence of subspaces. We also show that every frame of subspaces in a nite dimensional Hilbert space has frameness bound.

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithm to reconstruct a signal x from its noisy measurement F (x) +...

A common problem in applied mathematics is that of finding a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the Inverse Best Approximation Property (IBAP) if there exists a point that admits a...

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted $L^2$-space and sampling theorems entire functions Fock space dual uniform interpolating sequences. As consequence we obtain description of signal subspaces spanned by Hermite means Gabor frames.

We consider a generator F 1⁄4 ðf1; y ;fNÞ for either a multi-frame or a super-frame generated under the action of a projective unitary representation for a discrete countable group. Examples of such frames include Gabor multi-frames, Gabor super-frames and frames for shift-invariant subspaces. We show that there exists a unique normalized tight multi-frame (resp. super-frame) generator C 1⁄4 ðc...