Jayakumar Ramanathan

Linear independence of time-frequency translates

Abstract. The refinement equation φ(t) = ∑N2 k=N1 ck φ(2t − k) plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates |a|1/2φ(at − b) of φ ∈ L2(R), it is natural to ask if there exist similar dependencies among the time-frequency translates e2πibtf(t + a) of f ∈ L2(R). In other words, what is the effect of replacing the group representation of L2(R) induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection {(ak , bk)}k=1, the set of all functions f ∈ L2(R) such that {e2πibktf(t+ ak)}k=1 is independent is an open, dense subset of L2(R). It is conjectured that this set is all of L2(R) \ {0}.

On the wavelet transform of fractional Brownian motion

A theorem characterizing fractional Brownian motion by the covariance structure of its wavelet transform is established. The authors examine whether there are alternate Gaussian processes whose wavelet transforms have a natural covariance structure. In addition, the authors examine if there are any Gaussian processes whose wavelet transform is stationary with respect to the affine group (i.e. the statistics of the wavelet transform do not depend on translations and dilations of the process). >

Methods of Applied Fourier Analysis

Periodic functions hardy spaces prediction theory discrete systems and control theory harmonic analysis in Euclidean space distributions functions with restricted transforms phase space wavelet analysis the discrete Fourier transform the Hermite functions.

Asymptotic singular value decay of time-frequency localization operators

The Weyl correspondence is a convenient way to define a broad class of time-frequency localization operators. Given a region (Omega) in the time-frequency plane R2 and given an appropriate (mu) , the Weyl correspondence can be used to construct an operator L((Omega) ,(mu) ) which essentially localizes the time-frequency content of a signal on (Omega) . Different choices of (mu) provide different interpretations of localization. Empirically, each such localization operator has the following singular value structure: there are several singular values close to 1, followed by a sharp plunge in values, with a final asymptotic decay to zero. The exact quantification of these qualitative observations is known only for a few specific choices of (Omega) and (mu) . In this paper we announce a general result which bounds the asymptotic decay rate of the singular values of any L((Omega) ,(mu) ) in terms of integrals of (chi) (Omega ) * -(mu) 2 and ((chi) (Omega ) * -(mu) )^2 outside squares of increasing radius, where -(mu) (a,b) equals (mu) (-a, -b). More generally, this result applies to all operators L((sigma) ,(mu) ) allowing window function (sigma) in place of the characteristic functions (chi) (Omega ). We discuss the motivation and implications of this result. We also sketch the philosophy of proof, which involves the construction of an approximating operator through the technology of Gabor frames--overcomplete systems which allow basis-like expansions and Plancherel-like formulas, but which are not bases and are not orthogonal systems.

Time-frequency localization operators

A technique for producing signals whose energy is concentrated in a given region of the time-frequency plane is examined. The degree to which a given signal is concentrated in a region is measured by integrating its time-frequency distribution over the region. This method has been used for time-varying filtering. Localization operators based on the Wigner distribution and spectrogram are studied. Estimates for the eigenvalue decay and the smoothness and decay of the eigenfunctions are presented.<<ETX>>

Rigidity of minimal surfaces in S 3

Isometric deformations of compact minimal surfaces in the standard three-sphere are studied. It is shown that a given surface admits only finitely many noncongruent minimal immersions intoS3 with the same first fundamental form.

Time-frequency localization operators of Cohen’s class

A technique of producing signals whose energy is concentrated in a given region of the time-frequency plane is examined. The degree to which a particular signal is concentrated is measured by integrating a time-frequency distribution over the given region. This procedure was put forward by Flandrin, and has been used for time-varying filtering in the recent work of Hlawatsch, Kozek, and Krattenthaler. In this paper, the operators associated with the Wigner distribution and the spectrogram are considered. New results on the decay rate of the eigenvalues and the smoothness and decay of the eigenfunctions are presented.

Discrete Systems and Control Theory

Discrete time invariant linear systems are particularly amenable to the tools of Fourier analysis. Such systems, commonly used in engineering, are modeled by discrete convolution against a fixed sequence or filter. The Fourier series of this filter is the transfer function. Control theory begins with the observation that convolution against a filter cannot be physically realized if it involves knowledge of the future of a system. Filters that involve only past knowledge to implement are called causal. In many circumstances, the ideal filter turns out not to be causal. It is natural to ask for the causal filter that best fits these ideal criteria. In this section, we will treat one mathematical formulation of this type of problem, the solution of which will be an elegant application of the Hp theory developed in chapter 3.

The Existence and Structure of Rotational Systems in the Circle

By a rotational system, we mean a closed subset of the circle, , together with a continuous transformation with the requirements that the dynamical system be minimal and that respect the standard orientation of . We show that infinite rotational systems , with the property that map has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, . Because our main result makes no explicit mention of a global transformation on , we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation with finite preimages. In particular, there are no explicit conditions on the degree of . We then give a development of known results in the case where for an integer . The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.

Harmonic Analysis in Euclidean Space

The ideas that led to the representation of periodic functions as trigonometric series will be applied to functions defined on Rn. As before, we will attempt to expand functions in terms of characters parameterized, however, by Rn instead of Z. As a consequence, series expansions will be replaced by integral expansions, and the theory of inversion will have a different flavor than in the previous case. However, the underlying themes in this chapter will be the same as those of chapter 2: convolution and approximation, Plancherel’s theorem, Bochner’s theorem etc.