Multidimensional sampling theorems for multivariate discrete transforms

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

On Generalizations of Sampling Theorem and Stability Theorem in Shift-Invariant Subspaces of Lebesgue and Wiener Amalgam Spaces with Mixed-Norms

In this paper, we establish generalized sampling theorems, generalized stability theorems and new inequalities in the setting of shift-invariant subspaces of Lebesgue and Wiener amalgam spaces with mixed-norms. A convergence theorem of general iteration algorithms for sampling in some shift-invariant subspaces of Lp→(Rd) are also given.

The Effect of Training and Testing Process on Machine Learning in Biomedical Datasets

Training and testing process for the classification of biomedical datasets in machine learning is very important. The researcher should choose carefully the methods that should be used at every step. However, there are very few studies on method choices. The studies in the literature are generally theoretical. Besides, there is no useful model for how to select samples in the training and testing process. Therefore, there is a need for resources in machine learning that discuss the training and testing process in detail and offer new recommendations. This article provides a detailed analysis of the training and testing process in machine learning. The article has the following sections. The third section describes how to prepare the datasets. Four balanced datasets were used for the application. The fourth section describes the rate and how to select samples at the training and testing stage. The fundamental sampling theorem is the subject of statistics. It shows how to select samples. In this article, it has been proposed to use sampling methods in machine learning training and testing process. The fourth section covers the theoretic expression of four different sampling theorems. Besides, the results section has the results of the performance of sampling theorems. The fifth section describes the methods by which training and pretest features can be selected. In the study, three different classifiers control the performance. The results section describes how the results should be analyzed. Additionally, this article proposes performance evaluation methods to evaluate its results. This article examines the effect of the training and testing process on performance in machine learning in detail and proposes the use of sampling theorems for the training and testing process. According to the results, datasets, feature selection algorithms, classifiers, training, and test ratio are the criteria that directly affect performance. However, the methods of selecting samples at the training and testing stages are vital for the system to work correctly. In order to design a stable system, it is recommended that samples should be selected with a stratified systematic sampling theorem.

Sampling Theorems for Stochastic Signals. Appraisal of Paul L. Butzer’s Work

The sampling reconstruction theory is one of the great areas of the analysis in which Paul Leo Butzer earned longstanding and excellent theoretical results. Thus, we are forced either by earlier exhaustive presentations of his research activity and/or the highly voluminous material to restrict ourselves to a more narrow and precise sub-area in consideration; we discuss here, giving deeper insight, Paul Butzer’s sampling theoretical work with special attention concerning sampling stochastic signals.