Summability Techniques and Their Applications in Soft Computing

Summability Techniques and Their Applications in Soft Computing

Smita Sonker (National Institute of Technology Kurukshetra, India) and Alka Munjal (National Institute of Technology Kurukshetra, India)
DOI: 10.4018/978-1-5225-3035-0.ch013


Summability methods are a useful tool in dealing with the problems in the soft computing like in filtering of the signals and for stabilizing the systems. Signals can be in the form of various types of series (Infinite Series, Fourier series, etc.) and hence, summability theory is applicable in finding the error of approximation and degree of approximation of such signals. In this chapter, the authors gave an introductory discussion on summability theory and approximation of the signals. Further, they explained about the stability of the frequency response of the system. Also, they used the Fourier approximation in the soft computing models (multilayer perceptrons; radial basis function (RBF) or regularization networks, and fuzzy logic models) and found the output data of requirement.
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In this modern world, the soft computing becomes very effective in developing the functional and robust intelligent systems. The theory of the soft computing is developed by mathematical and logical frameworks considering the natural attributes in learning and reasoning. It enables the computing devices with more reasonable and logical for an effective and intelligent functioning. The soft computing exists in every area of our life which uses the technology like computer systems, transport vehicles, appliances, or the things with hidden digital electronics.

Summability techniques are very much applicable in soft computing as a tool to filter the signals in the form of series (Infinite Series, Fourier series, wavelets etc.) (Bachmann et al., 2012) using the various summability methods (Zygmund, 1988) such as matrix summability, Cesàro and Generalized Cesàro summability, Hölder summability, Harmonic summability, Riesz’s and Riesz’s typical means summability, Nörlund and Generalized Nörlund summability, indexed summability, Abel summability, Euler summability, Borel summability, Hausdorff summability, Banach summability etc. (Dutta and Rhoades, 2016). In this chapter, we will discuss the stability of the system using the summability and absolute summability methods.

Kumar et al. (2010) determined the new approach for Generating Parametric Orthogonal Wavelets in which scaling function filter represent as a product of two Laurent polynomials. In 2011, they used Discrete Wavelet Transform in place of Fourier transform and improve the results. Also, Kumar et al. (2016) determined the Wavelet Variance, Covariance and Correlation Analysis of BSE and NSE Indexes Financial Time Series. In 2016, Sonker and Munjal (2016a) determined a theorem on generalized absolute Cesàro summability with the help of sufficient conditions for infinite series and in (2016b), they used the concept of triangle matrices for obtaining the minimal set of sufficient conditions of infinite series to be bounded. In 2017, Sonker and Munjal (2017a) obtained boundness conditions of absolute summability factors. In this way by using the advanced summability method, we can improve the quality of the filters.

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