L 2 regularity of measurable solutions of a finite-difference equation of the circle
We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2. ::: ::: This work of Michel Herman (1942-2000) appeared only as a preprint of the Mathematics Institute, University of Warwick, dated May 1976. It was turned into TEX format by Claire Desescures. Minor editorial work was done by Albert Fathi.
The main objective of this study is to evaluate ball bearing defects under different operation conditions through vibration measurements. There are several tests conducted for healthy and defective bearing under variable speeds and load conditions. Experimental tests are conducted for six sets of ball bearings. Initially, a good bearing is fastened in the test rig and vibration signals are measured using the FFT analyzer to show the base-line performance of a healthy bearing. Then, the good bearing is exchanged by defective bearing and vibration signals are measured for each case separately under the same operation condition. Frequency domain, time domain and root-mean-square are used to describe various bearing defects. The experimental results are showed that each one of these methods is useful to identify the bearing problems. Also, the results proved that the significant variation in the root-mean-square at different rotational speed.