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This book publishes original research chapters on the theory of approximation by positive linear operators as well as theory of sequence spaces and illustrates their applications. Chapters are original and contributed by active researchers in the field of approximation theory and sequence spaces. Each chapter describes the problem of current importance and summarizes ways of their solution and possible applications which improve the current understanding pertaining to sequence spaces and approximation theory. The presentation of the articles is clear and self-contained throughout the book.
Each chapter describes the problem of current importance, and to summarize ways of their solution and possible applications, and to improve the current understanding pertaining to sequence spaces and approximation theory. The book consists of original fourteen research chapters. Corroborations of the main results are detailed and elegant. The list of chapters is arranged alphabetically by the last names of the first author of each chapter.
Chapter 1 is devoted to studying the geometric sequence spaces with geometric topology which is a nonlinear topology and explores salient features of the geometric topological sequence spaces. This chapter also discusses about normal topology and its properties, and introduces perfect, simple, symmetric geometric sequence spaces as well as duality between perfect geometric sequence spaces. Chapter 2 presents some results regarding the Fredholmnesss, compactness, and boundednes of composition operators defined on the second-order Cesaro function spaces and computes the composition operators’s essential norm. Chapter 3 introduces the concepts of deferred statistical convergence.
Chapter 4 obtains the asymptotic type theorem such as the Voronovskaya theorem, quantitative and Grüss–Voronovskaya type theorem. The order of approximation for the functions having a derivative equivalent with a function of bounded variation for these operators is presented. Chapter 5 introduces the Zachary space over R and finds that this is a Banach space of functions of bounded mean oscillation. Chapter 6 discusses some properties of Dunkl generalization of Szász operators via q-calculus by considering the new generalization of the power summability methods such as uniform convergence of this type of operators, Korovkin type theorem, Voronovskaya and Gruss–Voronovskaya type theorems, and obtains some results on weighted spaces of continuous functions. Chapter 7 is devoted to constructing the Szász–Jakimovski–Leviatan type operators by using sequence of nonnegative continuous functions. Chapter 8 obtains the approximation of signals belonging to Zygmund class associated with conjugate Fourier series and conjugate derived Fourier series. In Chapter 9, some effective semi-analytic and numerical methods are presented based on the following approaches. The first approach is a modified technique developed based on a concept of topology and perturbations theory which is named as modified homotopy perturbation method