We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e i|ξ | α , where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ | −δ sin(|ξ | α ) for 0 δ α.

By using tools of time‐frequency analysis, we obtain some improved local well‐posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces ℳ0,sp,1.

We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distributions of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of special classes of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.

Abstract. We prove that for symbols in the modulation spaces M p,q , p ≥ q, the associated multilinear pseudodifferential operators are bounded on products of appropriate modulation spaces. In particular, the symbols we study here are defined without any reference to smoothness, but rather in terms of their time-frequency behavior.

Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.

A finite collection of unit vectors S ⊂ R n is called a spherical two-distance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We prove that if a = −b, then a two-distance set that forms a tight frame for R n is a spherical embedding of a strongly regular graph. We also describe all two-distance tight frames obtained from a given graph. Together with an earlier work by S. Waldron (2009) [22] on the equiangular case, this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all twodistance 2-designs.

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We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on L p spaces. The main tools are Gabor frames and methods from time-frequency analysis.