No abstract

Discrete affine systems are obtained by applying dilations to a given shiftinvariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are``global'' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions.

Discrete a ne systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the a ne system is due, rst and foremost, to the fact that it is not invariant under shifts. A ne frames carry the additional di culty that they are \global" in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the a ne system with the aid of two new notions: the a ne product, and a quasi-a ne system. This leads to a characterization of a ne frames; the induced characterization of tight a ne frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the a ne product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple su cient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom o ered by redundancy can be fully exploited in these constructions.

A simple characterization is given of finitely generated subspaces of L 2 (IRd) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for 'local' spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces.As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years' standing.

Abstract--Identifying anomalies rapidly and accurately is critical to the efficient operation of large computer networks. Accurately characterizing important classes of anomalies greatly facilitates their identification; however, the subtleties and complexities of anomalous traffic can easily confound this process. In this paper we report results of signal analysis of four classes of network traffic anomalies: outages, flash crowds, attacks and measurement failures. Data for this study consists of IP flow and SNMP measurements collected over a six month period at the border router of a large university. Our results show that wavelet filters are quite effective at exposing the details of both ambient and anomalous traffic. Specifically, we show that a pseudo-spline filter tuned at specific aggregation levels will expose distinct characteristics of each class of anomaly. We show that an effective way of exposing anomalies is via the detection of a sharp increase in the local variance of the filtered data. We evaluate traffic anomaly signals at different points within a network based on topological distance from the anomaly source or destination. We show that anomalies can be exposed effectively even when aggregated with a large amount of additional traffic. We also compare the difference between the same traffic anomaly signals as seen in SNMP and IP flow data, and show that the more coarse-grained SNMP data can also be used to expose anomalies effectively.