Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...

We compare three types of coherent Riesz families (Gabor systems, Wilson bases, and wavelets) with respect to their perturbation stability under convolution with elements of a family of typical channel functions. This problem is of key relevance in the design of modulation signal sets for digital communication over time–invariant channels. Upper and lower bounds on the orthogonal perturbation a...

Repeated convolution of a probability measure on Z leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomée which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomée). Using Fourier transform argument...

We consider the convolution model where i.i.d. random variables Xi having unknown density f are observed with additive i.i.d. noise, independent of the X’s. We assume that the density f belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the ...

Given reasonably chosen fixed sets of reproduction letters and/or their probabilities, the authors define new rate-distortion functions which are coding bounds under these alphabet constraints. Calculations are made of these functions for the Gaussian and Laplacian sources and the squared-error distortion measure and find that performance near the rate-distortion bound is achievable using a rep...

We consider the problem of distributed lossy linear function computation in a tree network. We examine two cases: (i) data aggregation (only one sink node computes) and (ii) consensus (all nodes compute the same function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network computation rate as a function of incremental distortion...

Rate-distortion theory is considered for the Shannon cipher system (SCS). The admissible region of cryptogram rateR, key rate Rk, legitimate receiver’s distortion D, and wiretapper’s uncertainty h is determined for the SCS with a noisy channel. Furthermore, inner and outer bounds of the admissible region of R, Rk, D, and wiretapper’s attainable minimum distortion ~ D are derived for the SCS wit...

Waveform coding of audio signals at low bit rates generally results in coding errors. In high-quality applications these must remain inaudible. The bit rate required to code audio signals without audible errors depends on both the signal's power spectral density function and masking properties of the human ear. It is shown how rate distortion theory and psy-choacoustic models of hearing can be ...

We present a global Ziv-Zakai-type lower bound on the mean square error for estimation of signal parameter vectors, where some components of the distortion function may be periodic. Periodic distortion functions arise naturally in the context of direction of arrival or phase estimation problems. The bound is applied to an image registration problem, and compared to the performance of the Maximu...

We show time lower bounds for both online integer multiplication and convolution in the cell-probe model with word size w. For the multiplication problem, one pair of digits, each from one of two n digit numbers that are to be multiplied, is given as input at step i. The online algorithm outputs a single new digit from the product of the numbers before step i+ 1. We give a lower bound of Ω( δ w...