174 research outputs found
Wavelet Electrodynamics I
A new representation for solutions of Maxwell's equations is derived. Instead
of being expanded in plane waves, the solutions are given as linear
superpositions of spherical wavelets dynamically adapted to the Maxwell field
and well-localized in space at the initial time. The wavelet representation of
a solution is analogous to its Fourier representation, but has the advantage of
being local. It is closely related to the relativistic coherent-state
representations for the Klein-Gordon and Dirac fields developed in earlier
work.Comment: 8 Pages in Plain Te
Splines in Compressed Sensing
It is well understood that in any data acquisition system reduction in the amount of data reduces the time and energy, but the major trade-off here is the quality of outcome normally, lesser the amount of data sensed, lower the quality. Compressed Sensing (CS) allows a solution, for sampling below the Nyquist rate. The challenging problem of increasing the reconstruction quality with less number of samples from an unprocessed data set is addressed here by the use of representative coordinate selected from different orders of splines. We have made a detailed comparison with 10 orthogonal and 6 biorthogonal wavelets with two sets of data from MIT Arrhythmia database and our results prove that the Spline coordinates work better than the wavelets. The generation of two new types of splines such as exponential and double exponential are also briefed here .We believe that this is one of the very first attempts made in Compressed Sensing based ECG reconstruction problems using raw data.
Biorthogonal-wavelet-based method for numerical solution of volterra integral equations
© 2019 by the authors. Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is converted into matrix equations via these generators. We present examples of the generated pseudo-splines quasi-affine tight framelet systems. Some numerical results to validate the proposed method are presented to illustrate the efficiency and accuracy of the method
Asymptotic Normality of Quadratic Estimators
We prove conditional asymptotic normality of a class of quadratic
U-statistics that are dominated by their degenerate second order part and have
kernels that change with the number of observations. These statistics arise in
the construction of estimators in high-dimensional semi- and non-parametric
models, and in the construction of nonparametric confidence sets. This is
illustrated by estimation of the integral of a square of a density or
regression function, and estimation of the mean response with missing data. We
show that estimators are asymptotically normal even in the case that the rate
is slower than the square root of the observations
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