Fourier Transform

The application of Fourier transform (FT) in signal processing and physical sciences has increased in the past decades. Almost all the textbooks on signal processing or physics have a section devoted to the FT theory. For this reason, this book focuses on signal processing and physical sciences. The book chapters are related to fast hybrid recursive FT based on Jacket matrix, acquisition algorithm for global navigation satellite system, determining the sensitivity of output parameters based on FFT, convergence of integrals of products based on Riemann-Lebesgue Lemma function, extending the real and complex number fields for treating the FT, nonmaterial structure, Gabor transform, and chalcopyrite bioleaching. The book provides applications oriented to signal processing and physics written primarily for engineers, mathematicians, physicians and graduate students, will also find it useful as a reference for their research activities

Certain fundamental properties of generalized natural transform in generalized spaces

This paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386-1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed

Exceptional Lebesgue densities and random Riemann sums

We will examine two topics in this thesis. Firstly we give a result which improved a bound for a question asking which values the Lebesgue density of a measurable set in the real line must have (joint work with Toby O'Neil and Marianna Csörnyei). We also show how this result relates to the results obtained by others. Secondly, we give several results which indicate when a Lebesgue measurable function has a random Riemann integral which converges, in either the weak and strong sense. A Lebesgue measurable set A, subset of R, has density either 0 or 1 at almost every point. Here the density at some point x refers to the proportion of a small ball around x which belongs to A, in the limit as the size of the ball tends to 0. Suppose that A is not either a nullset, which has density 0 at every single point, or the complement of a nullset, which similarly has density 1 everywhere. Then there are certain restrictions on the range of possible values at those exceptional points where the density is neither 0 nor 1. In particular, it is now known that if δ < 0.268486 ..., where the exact value is the positive root of 8δ^3+8 δ^2+ δ1 = 0, then there must exist a point at which the density of A is between δ and 1 - δ, and that this does not remain true for any larger value of δ. This was proved in a recent paper by Ondrej Kurka. Previous to his work our result given in this thesis was the best known counterexample. We give the background to this, construct the counterexample, and discuss Kurka's proof of the exact bound. The random Riemann integral is defined as follows. Given a Lebesgue measurable function f : [0, 1] \rightarrow R and a partition of [0, 1] into disjoint intervals, we can choose a point belonging to each interval, independently and uniformly with respect to Lebesgue measure. We then use these random points to form a Riemann sum, which is itself a random variable. We are interested in knowing whether or not this random Riemann sum converges in probability to some real number. Convergence in probability to r means that the probability that Riemann sum differs from r by more than \varepsilon, is less than \varepsilon, provided that the maximum length of an interval in the partition is sufficiently small. We have previously shown that this type of convergence does take place provided that f is Lebesgue integrable. In other words, the random Riemann integral, defined as the limit in probability of the random Riemann sums, has at least the power of the Lebesgue integral. Here we prove that the random Riemann integral of f does not converge unless |f|^1-e is integrable for e > 0 arbitrarily small. We also give another, more technical, necessary condition which applies to functions which are not Lebesgue integrable but are improper Riemann integrable. We have also done some work on the question of almost sure convergence. This works slightly differently. We must choose, in advance, a sequence of partitions (Pn)n∞=1, with the size of the intervals of Pn tending to zero. We form a probability space on which we can take random Riemann sums independently on each partition of the sequence. Almost sure convergence means that the sequence of random Riemann sums converges to some (unique) limit with probability 1 in this space. There are two complementary results; firstly that almost sure convergence holds if the function is in Lp and the sequence of partition sizes is in l^p-1 for some p \ge 1. Secondly, we have a partial converse which only applies to nonnegative functions, and if the ratio between the lengths smallest and biggest intervals in each partition is bounded uniformly. This says that if for some p \ge 1 f is not in L^p and the partition sizes are not in l^p-1, then the sequence of Riemann sums diverges with probability 1

New Trends on Nonlocal and Functional Boundary Value Problems

In the last decades, boundary value problems with nonlocal and functional boundary conditions have become a rapidly growing area of research. The study of this type of problems not only has a theoretical interest that includes a huge variety of differential, integrodifferential, and abstract equations, but also is motivated by the fact that these problems can be used as a model for several phenomena in engineering, physics, and life sciences that standard boundary conditions cannot describe. In this framework, fall problems with feedback controls, such as the steady states of a thermostat, where a controller at one of its ends adds or removes heat depending upon the temperature registered in another point, or phenomena with functional dependence in the equation and/or in the boundary conditions, with delays or advances, maximum or minimum arguments, such as beams where the maximum (minimum) of the deflection is attained in some interior or endpoint of the beam. Topological and functional analysis tools, for example, degree theory, fixed point theorems, or variational principles, have played a key role in the developing of this subject. This volume contains a variety of contributions within this area of research. The articles deal with second and higher order boundary value problems with nonlocal and functional conditions for ordinary, impulsive, partial, and fractional differential equations on bounded and unbounded domains. In the contributions, existence, uniqueness, and asymptotic behaviour of solutions are considered by using several methods as fixed point theorems, spectral analysis, and oscillation theory