arXiv subjects

Eric Weber

Publications and source records attributed to Eric Weber.

27 records · Page 2Linked to original sources

arXiv2004

Encryption Schemes using Finite Frames and Hadamard Arrays

We propose a cipher similar to the One Time Pad and McEliece cipher based on a subband coding scheme. The encoding process is an approximation to the One Time Pad encryption scheme. We present results of numerical experiments which suggest that a brute force attack to the proposed scheme does not result in all possible plaintexts, as the One Time Pad does, but still the brute force attack does not compromise the system. However, we demonstrate that the cipher is vulnerable to a chosen-plaintext attack.

cs.CR↗

arXiv2003

Orthogonal Frames of Translates

Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in $\ltwod$. The characterizations are applied to Bessel sequences which have an affine structure, and a quasi-affine structure. These also lead to characterizations of superframes. Moreover, we characterize perfect reconstruction, i.e. duality, of subspace frames for translation invariant (bandlimited) subspaces of $\ltwod$.

math.FA↗

arXiv2003

Geometric Aspects of Frame Representations of Abelian Groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

math.FA↗

arXiv2003

The Geometry of Sampling on Unions of Lattices

In this short note we show two results concerning sampling translation invariant subspaces of $\ltwod$ on unions of lattices. The first result shows that the sampling transform on a union of lattices is a constant times an isometry if and only if the sampling transform on each individual lattice is so. The second result demonstrates that the sampling transforms of two unions of lattices on two bands have orthogonal ranges if and only if correspondingly the sampling transforms of each pair of lattices have orthogonal ranges. We then consider sampling on shifted lattices.

math.FA↗

arXiv2003

Ellipsoidal tight frames and projection decompositions of operators

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H, and we analyze the set of attainable frame bounds. In the case where H is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite-dimensional case is a result we have proven which concerns the decomposition of a positive invertible operator into a strongly converging sum of (not necessarily mutually orthogonal) self-adjoint projections. This decomposition result implies the existence of tight frames in the ellipsoidal surface determined by the positive operator. In the real or complex finite dimensional case, this provides an alternate (but not algorithmic) proof that every such surface contains tight frames with every prescribed length at least as large as dim H. A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.

math.OA↗

arXiv2001

Robertson Type Theorems for Frames

We extend Robertson's theorem to apply to frames generated by the action of a discrete, countable abelian unitary group. Within this setup we use Stone's theorem and the theory of spectral multiplicity to analyze wandering frame collections. Motivated by wavelet theory, we explicitly apply our results to the action of the integers given by translations on $\ltwo$. This yields a new functional analytic method of constructing a wavelet from a multiresolution analysis.

math.FA↗

arXiv2000

Wavelets with the Translation Invariance Property of Order N

All wavelets can be associated to a multiresolution like structure, i.e. an incr easing sequence of subspaces of L^2(R). We consider the interaction of a wavel et and the translation operator in terms of which of the subspaces in this multi resolution like structure are invariant under the translation operator. This ac tion defines the notion of the translation invariance property of order n. In this paper we show that wavelets of all levels of translation invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral di lation factors.

math.FA↗

arXiv1999

Applications of the Wavelet Multiplicity Function

This paper examines the wavelet multiplicity function. An explicit formula for the multiplicity function is derived. An application to operator interpolation is then presented. We conclude with several remarks regarding the wavelet connectivity problem.

math.FA↗

arXiv1999

On the Translation Invariance of Wavelet Subspaces

An examination of the translation invariance of $V_0$ under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.

math.FA↗