A sharp L p spectral multiplier theorem of Mihlin-Hörmander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.2000 Mathematics Subject Classification. Primary: 42B15, 43A85; Secondary: 53C26.

In this paper we propose a novel machine-learning method for anomaly detection applicable to data with periodic characteristics where randomly varying period lengths are explicitly allowed. A multi-dimensional time series analysis is conducted by training a data-adapted classifier consisting of deep convolutional neural networks performing phase classification. The entire algorithm including data pre-processing, period detection, segmentation, and even dynamic adjustment of the neural networks is implemented for fully automatic execution. The proposed method is evaluated on three example datasets from the areas of cardiology, intrusion detection, and signal processing, presenting reasonable performance.

In this paper, a series of convolutional-type predictive neural networks are proposed for the issue of fading channel forecasting for orthogonal frequency-division multiplexing (OFDM) transmission systems in a multiple-input and multiple-output (MIMO) mode via a noisy channel. The proposed neural networks all employ convolutional connections that operate in a translation-invariant manner in the frequency domain of the time-varying channel transfer function, which effectively tackles the essential challenges of high dimensionality and denoising. Each of the proposed convolutional-type neural networks is built on a specific overall network architecture and functions as an independent predictor that offers advantages regarding a specific aspect such as accuracy over a certain prediction span or computational effort. Comparative evaluations against common prediction methods such as the Kalman filtering scheme and the standard long-short term memory units (LSTMs) are provided on the basis of transmission simulations over dispersive fading channels with Rayleigh components according to the well-established 3GPP Long Term Evolution (LTE) standards. INDEX TERMS Wireless communications, deep learning, convolutional neural networks, LSTMs, time series analysis, fading channel forecasting, MIMO.