By Mitsuo Morimoto

This e-book treats round harmonic enlargement of actual analytic features and hyperfunctions at the sphere. simply because a one-dimensional sphere is a circle, the easiest instance of the idea is that of Fourier sequence of periodic features. the writer first introduces a method of complicated neighborhoods of the field through the Lie norm. He then reviews holomorphic services and analytic functionals at the complicated sphere. within the one-dimensional case, this corresponds to the research of holomorphic capabilities and analytic functionals at the annular set within the complicated aircraft, hoping on the Laurent sequence enlargement. during this quantity, it really is proven that an identical notion nonetheless works in a higher-dimensional sphere. The Fourier-Borel transformation of analytic functionals at the sphere can be tested; the eigenfunction of the Laplacian may be studied during this manner.

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Extra resources for Analytic Functionals on the Sphere

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In Chapter 2 an in-depth analysis of neural networks is presented. Wavelet Neural Networks Wavelet networks are a new class of networks that combine classic sigmoid neural networks and wavelet analysis. Wavelet networks have been used with great success in a wide range of applications. Wavelet analysis has proved to be a valuable tool for 20 MACHINE LEARNING AND FINANCIAL ENGINEERING analyzing a wide range of time series and has already been used with success in image processing, signal denoising, density estimation, signal and image compression, and time-scale decomposition.