A generalisation of a theorem of Pati on the Nörlund summability of Fourier series

Pati generalised a theorem by Siddiqi on the harmonic summability of Fourier series, by replacing the special sequence by a more gererral sequence. Dikshit proved an analogue of it in the case of conjugate series of Fourier series and further he improved his theorem by introducing a functional factor. The problem which was unsolved as to the same improvement and generalisation can be done in case of Fourier series is tackled and proved by me in the paper attached herewith.     

Joseph Fourier's Anticipation of Linear Programming

The name of Joseph Fourier (1768-1830) is largely associated with the mathematical analysis of heat diffusion and methods of solving partial differential equations by means of Fourier series, Fourier integrals and the calculus of differential operators. But his interests in mathematics encompassed other fields also, and one of his achievements was to create singlehanded a basic theory of linear programming.     

On the absolute Cesàro summability of Fourier series of functions of Lebesgue classL v and some related problems in the theory of Fourier constants

Summary The author studies certain aspects of a problem on Fourier constants which emerges out of his attempt at proving a recent result ofTsuchikura on the absolute Cesàro summability of Fourier series by means of a technique that exploits properties of Fourier costants. He proves,inter alia, the Fourier-power series analogue of aconjecture on Fourier constants, which he has raised in this paper.     

Lanczos-Like σ-Factors for Reducing the Gibbs Phenomenon in General Orthogonal Expansions and Other Representations

The first attempt for reducing the Gibbs phenomenon in an orthogonalexpansion, besides the usual one of Fourier series, is due to Cooke in1927–1928 for the Fourier Bessel series.However, his work was limited tothe well-known Fejer averaging of the series. For the past 10 years or so,we have tried a parallel to the more effective Lanczos-type localaveraging method of the Fourier series. As expected, such efforts werehindered by the lack of realizable tools for the general orthogonalexpansion that parallels the familiar simple ones of the Fourier seriesand transforms. During the past 3 years, we have succeeded in developinga simple direct method of filtering the Gibbs phenomenon in Fourier--Besselseries, Hankel transforms representation, and a number of orthogonalpolynomials series expansions. This parallels the equivalent result ofLanczos, which he obtained for his local averaging with the help ofthe (Fourier) convolution theorem.     

Eisenstein series on quaternion half-space

Kaufhold [5] calculated the Fourier coefficients of the Siegel's Eisenstein series of degree 2 and obtained its analytic continuation and functional equation. In this paper, we follow his procedure to obtain the analytic continuation and a functional equation of the Eisenstein series on quaternion half-space defined by Krieg [7]. S. Nagaoka has announced a similar result (see below). The author wishes to express his gratitude to Prof. Walter Baily Jr., for suggesting the topic of this research and for his encourgement and advice and to Dr. A. Krieg and the referee for helpful suggestions and corrections.     

Engaging a third-grade student with autism spectrum disorder in an error finding activity

This paper describes a case study of one mainstreamed third grade student with autism spectrum disorder (ASD) and his ability to explain his solutions for two-digit addition problems, and find and explain the mistake when presented with incorrectly solved addition problems. The study is presented as a counterexample to deficit views of ASD, views that focus on lack of communication skills, not being able to see someone else’s point of view, and poor executive functions. Each encounter with the student is analyzed in two ways, first analyzing his mathematical knowledge, and then analyzing obstacles the student faces that are associated with ASD. Some obstacles are overcome by the student on his own and others are overcome with the help of the researcher, who responds to the student’s thinking, and supports his endeavor to engage with a challenging activity.     

Localization in bounded summation by Cesaro,Riesz, and Abel methods for multiple Fourier series

We establish the conditions under which the localization principle is valid in bounded summation by Cesaro, Riesz, and Abel methods for multiple Fourier series of functions of Liouville classes.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 355–364, October, 1972.The author wishes to use this occasion to express his deep gratitude to his scientific director Professor V. A. Il'in for formulating the problem and for constant help with the paper, and also Sh. Alimov for his attention to the paper.     

Developing the concept of linear function: one student’s experiences with dynamic physical models

Developing a view of functions as systematic processes involving co-variation among variables has been identified as a goal for mathematics learners at the pre-university level. In this investigation I examined the processes used by an eighth-grade student to interpret linear functions originating in dynamic physical models and the processed he used to link his interpretations to tables, equations, and graphs. The student deepened his understanding of functions by generalizing his view of multiplication beyond that of products resulting from the multiplication of individual factors or from repeated addition. He was enabled to do this by building links between graphic and tabular representations of the functions generated from his exploration with dynamic physical models and by comparing tables of different linear functions. This paper suggests that the development of a student’s reasoning about functions originating in dynamic physical models can be interpreted in terms of generalized multiplicative processes that may occur thorough mapping variations and that a student who interprets such functions as generalized multiplicative processes may use notational variations to generate representing equations.     

Fourier Series and Approximation on Hexagonal and Triangular Domains

Several problems on Fourier series and trigonometric approximation on regular hexagonal and triangular domains are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation, and best approximation by trigonometric functions with both direct and inverse theorems. One of the objectives of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that permits an extensive theory for Fourier series and approximation.     

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter’s Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini’s type pointwise convergence theorem are proved. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Looking back to the roots of partially correct constructs: The case of the area model in probability

We use the notion Partially Correct Constructs (PaCCs) for students’ constructs that partially match the mathematical principles underlying the learning context. A frequent expression of partial construction of mathematical principles is that a student’s words or actions provide an inaccurate or misleading picture of the student’s knowledge. In this study, we analyze the learning process of a grade 8 student, who learns a topic in elementary probability. The student successfully accomplishes a sequence of several tasks without apparent difficulty. When working on a further task, which seems to require nothing beyond his proven competencies, he encounters difficulties. Using the epistemic actions of the RBC model for abstraction in context as tracers, we analyze his knowledge constructing processes while working on the previous tasks, and identify some of his constructs as PaCCs that are concealed in these processes and explain his later difficulties. In addition, our research points to the complexity of the knowledge structures students are expected to deal with in their attempts to learn an elementary mathematical topic with understanding.     

Hidden lemmas in the early history of infinite series

Euler, Fourier, Poisson and Cauchy appear to have used, in a more or less implicit form, some facts on infinitely small quantities. Attempting to state and prove several lemmata, I shall discuss their relationships to interchanges of limits in series and integrals. Early methods of summation for divergent series and integrals, including a conjecture of Poisson, are discussed.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

A connection between inclusion theorems and the uniform convergence of multiple Fourier series

A relation is established between inclusion theorems and the uniform convergence of Fourier series for the case of functions of many variables. The one-dimensional case has already been analyzed by Ul'yanov.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 139–148, August, 1972.The author wishes to express his gratitude to P. L.Ul'yanov under whose guidance this paperwas prepared.     

Absolutely convergent expansions

The classical Bernstein estimate for absolutely convergent Fourier series and its multivariate extension are improved. Analogous results for absolutely convergent expansion by Legendre polynomials and by spherical harmonic polynomials are given. For expansion by a Fourier series it follows that the present result implies the classical result. Moreover, there are many functions that are shown to have an absolutely convergent Fourier series using the present estimate while not satisfying the conditions of the previous one.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.     

Generalized Hardy identity and relations to Gibbs-Wilbraham and Pinsky phenomena

We consider the Fourier series of the indicator functions of several dimensional balls. For the spherical partial sum of the Fourier series, we extract the Gibbs-Wilbraham (or Gibbs), Pinsky and the third phenomena as an extension of Hardy's identity. The third phenomenon has been shown by Kuratsubo recently. We prove the Gibbs-Wilbraham phenomenon for all dimensions and give another proof of the Pinsky phenomenon. Pinsky gave the order of the divergence for the Fourier inversion at the origin. We give the order of the divergence of the Fourier series at the origin and show that both orders coincide. We also investigate the uniform convergence for the Fourier series and the Fourier inversion.     

Algorithmically random series and Brownian motion

We consider some random series parametrised by Martin-Löf random sequences. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.     

Fourier series representation of the coupler curves of spatial linkages

In this paper, the three-dimensional Fourier series which can describe the coupler curves of spatial linkages are derived. Based on the geometric properties of one-dimensional and two-dimensional Fourier series, a geometric constraint conditions are proposed to establish the three-dimensional Fourier series. The geometric meaning of mechanism for the three-dimensional Fourier series is put forward. Finally, an example is given to show the feasibility and the validity of this approach.     

On the spectrum of periodic signals

In theory, there are many methods for the representation of signals. In practice, however, Fourier analysis involving the resolution of signals into sinusoidal components is used widely. There are several methods for Fourier analysis available for representation of signals. If the signal is periodic, then the Fourier series is used to represent the signal in terms of a set of harmonically related sinewaves. In this article new formulae for evaluating the trigonometric Fourier series coefficients when the signal under consideration is polynomial are developed by changing the integration to a derivation form. The solution presented here is an extension of the formulae proposed by Al-Smadi and Wilkes to the trigonometric Fourier series. The proposed technique is a powerful tool that can be used for solving practical science and engineering problems without excessive tedium.     

Théorèmes du type Ingham et application à la théorie du contrôle

Ingham [4] improved a previous result of Wiener [10] on nonharmonic Fourier series. Modifying his weight function we obtain optimal results improving several earlier theorems of Kahane [7], Castro and Zuazua [2] and of Jaffard, Tucsnak, and Zuazua [5]. Then we apply these results to simultaneous observability problems.