Gibbs phenomenon removal by adding Heaviside functions

We define a kind of spectral series to filter off completely the Gibbs phenomenon without overshooting and distortional approximation near a point of discontinuity. The construction of this series is based on the method of adding the Fourier coefficients of a Heaviside function to the given Fourier partial sums. More precisely, we prove the uniform convergence of the proposed series on the class of piecewise smooth functions. Also, we attach two numerical examples which illustrate the uniform convergence of the suggested series in comparison with the Fourier partial sums.     

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

Convergence and formal manipulation in the theory of series from 1730 to 1815

In the early calculus mathematicians used convergent series to represent geometrical quantities and solve geometrical problems. However, series were also manipulated formally using procedures that were the infinitary extension of finite procedures. By the 1720s results were being published that could not be reduced to the original conceptions of convergence and geometrical representation. This situation led Euler to develop explicitly a more formal approach which generalized the early theory. Formal analysis, which was predominant during the second half of the 18th century despite criticisms of it by some researchers, contributed to the enlargement of mathematics and even led to a new branch of analysis: the calculus of operations. However, formal methods could not give an adequate treatment of trigonometric series and series that were not the expansions of elementary functions. The need to use trigonometric series and introduce nonelementary functions led Fourier and Gauss to reject the formal concept of series and adopt a different, purely quantitative notion of series.     

Examples of divergent fourier series for classes of functions of bounded Λ-variation

The author has shown earlier that the requirement that a continuous function belong to the class HBV ([?π, π] ^m ) for m ≥ 3 is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.     

An Observation on Two Methods of Obtaining Solutions to Variational Problems

In this paper, we compare two methods for obtaining solutions for free problems in the calculus of variations. The first is due to Carathéodory (Ref. 1) and the second due to Leitmann (Ref. 2). Both methods introduce the notion of equivalent variational problems. Using either approach, an auxiliary problem is obtained for which the solution is more easily obtained. We compare both approaches by using each to solve the same class of examples. We conclude our discussion by unifying the two approaches into one and illustrating the potential of this new method through the use of an elementary example.     

Variational calculus with sums of elementary tensors of fixed rank

In this article we introduce a calculus of variations for sums of elementary tensors and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the solution of a linear system in structured tensor format. Moreover, we discuss the solution of an eigenvalue problem with sums of elementary tensors. This example can be viewed as a prototype of a constrained minimization problem. For the numerical treatment, we suggest a method which has the same order of complexity as the popular alternating least square algorithm and demonstrate the rate of convergence in numerical tests.     

The Peano-Baker series

This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually treated as a folklore result or as a purely formal tool. Here, a simple convergence result is given, and two examples illustrate that the series can be used explicitly as well.     

Sobolev spaces and symbolic calculus

We use elementary theory of distributions and geometry of Euclidean spaces to obtain a theorem on the symbolic calculus of several variables in spaces of Fourier series with weights.     

Generalized Fourier transform on an arbitrary triangular domain

In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach. We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions. The orthogonality and completeness of the systems are then proved. Some essential convergence properties of the generalized Fourier series are discussed. Error estimates are obtained in Sobolev norms. Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated. This work was supported by the Major Basic Project of China (No. G19990328) and National Natural Science Foundation of China (No. 60173021).     

A Critical Survey of Queueing Theory — Part 2

In this part of the paper, an elementary method (Erlang's) of setting up equations which describe queueing situations is discussed. Methods of solving these equations are given by reference to a simple example, and then to a more complex one.     

Gibbs Phenomenon in Wavelet Analysis

We prove Dirichlet-type pointwise convergence theorems for the wavelet transforms and series of discontinuous functions and we examine the Gibbs ripples close to the jump location. Examples are given of wavelets without ripples, and an example (the Mexican hat) shows that the Gibbs ripple in continuous wavelet analysis can be 3.54% instead of 8.9% of the Fourier case. For the discrete case we show that there exist two Meyer type wavelets the first one has maximum ripple 3.58% and the second 9.8%. Moreover we describe several examples and methods for estimating Gibbs ripples both in continuous and discrete cases. Finally we discuss how a wavelet transform generates a summability method for the Fourier case.     

On properties of the iterative maximum likelihood reconstruction method

In this paper, we continue our investigations⁶ on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback–Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementary proof of convergence of the iterative method is given for the discrete case. Numerical examples have already been given in References 6. Here, an example is considered which can be worked out analytically and which demonstrates fundamental properties of the algorithm.     

Evaluating views of lecturers on the consistency of teaching content with teaching approach: traditional versus reform calculus

This study aims to evaluate the consistency of teaching content with teaching approaches in calculus on the basis of lecturers' views. In this sense, the structures of the examples given in two commonly used calculus textbooks, both in traditional and reform classrooms, are compared. The content analysis findings show that the examples in both textbooks are presented in a rather formal language and generally highlight procedural knowledge. And, even though the examples in the reform book chosen are structured using multiple representations, only a small number of them incorporated the usage of instructional technology. The lecturers' views which were gathered indicated that, although, on the one hand, the example structures of the traditional textbook largely overlapped with the characteristics of the traditional approach, the example structures of the reform textbook, on the other hand, were found to be inconsistent with the characteristics of the reform approach, especially with regard to its environment and knowledge components. At the end of the paper, some suggestions for further studies are provided for book authors and researchers.     

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejér on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.     

牛曼─贝塞尔级数的共轭级数

本文讨论了牛曼─贝塞尔级数的共轭级数，建立了其部分和与相应的共轭Ｆｏｕｒｉｅｒ三角级数的部分和之间的关系，同时给出了两个收敛定理。     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

The evaluation of double series

A nonlinear transformation whose domain is the class of Pringsheim convergent double series is introduced, defined, and studied for the purpose of accelerating the convergence of a double series. A number of theorems are established which enable one to make effective use of the transformation. In particular, special attention is given to the case in which the transformed series converges uniformly better than the original double series. Finally, several examples are given to illustrate the effectiveness of the transform.     

Stationary fields with positive angle

An analytic characterization of stationary fields with a positive angle between the vertical past and future, in the sense of Helson and Lowdenslager, is given. The connection between the positivity of the angle and the convergence of the Fourier series for certain functions is studied. The case where both vertical and horizontal angles are positive is discussed. An example showing the positivity of the vertical angle while the horizontal angle is zero is provided. Proofs depend on various characterizations of densities admitting a positive angle for the circle case.     

Fourier series optimization opportunity

This note discusses the introduction of Fourier series as an immediate application of optimization of a function of more than one variable. Specifically, it is shown how the study of Fourier series can be motivated to enrich a multivariable calculus class. This is done through discovery learning and use of technology wherein students build the sine Fourier series for the simple function f(x)?=?x and then generalize to the nth term sine Fourier series for a general function, f(x). It is shown how the students can then explore the power of the Fourier series to represent functions.     

An Unconditionally Stable Difference Approximation for a Class of Nonlinear Dispersive Equations

An unconditionally stable leapfrog finite difference scheme for a class of nonlinear dispersive equations is presented and analyzed. The solvability of the difference equation which is a tridiagonal circular linear system is discussed. Moreover, the convergence and stability of the difference scheme are also investigated by a standard argument so that more difficult priori estimations are avoided. Finally, numerical examples are given.