Basic Trigonometric Korovkin Approximation for Fuzzy Valued Functions of Two Variables

In the first part of our theses we give sufficient conditions for the absolute convergence of the double Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to double Fourier series. In the second part we give sufficient conditions for the absolute convergence of the double Walsh-Fourier series of a function. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity and s-bounded fluctuation.

This paper is concerned with the convergence of double trigonometric series and Fourier series. Since the beginning of the 20th century, many authors have studied on those series. Also, Ferenc <TEX>$M{\acute{o}}ricz$</TEX> has studied the convergence of double trigonometric series and double Fourier series so far. We consider <TEX>$L^p(T^2)$</TEX>-convergence results focused on the Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> studies from the second half of the 20th century up to now. In section 2, we reintroduce some of Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> remarkable theorems. Also we investigate his several important results. In conclusion, we investigate his research trends and the simple minor genealogy from J. B. Joseph Fourier to Ferenc <TEX>$M{\acute{o}}ricz$</TEX>. In addition, we present the research minor lineage of his study on <TEX>$L^p(T^2)$</TEX>-convergence.

Jurkat and Peyerimhoff have characterized monotone Fouriereffective summability methods as those which are stronger than logarithmic order summability. Here the analogous result for double Fourier series is obtained assuming unrestricted rectangular convergence. It is also shown that there is a class of order summability methods, which are weaker than any Cesà ro method, for which the double Fourier series of any $f \in L$ is restrictedly summable almost everywhere. Finally, it is shown that square logarithmic order summability has the localization property for exponentially integrable functions.

A computationally useful expression for the modified Green' function in a rectangular region is derived. This can be used to obtain solutions of Laplace' equation or Poisson' equation for the electrical potential associated with any configuration of current sources that satisfies the condition of charge balance, and represents a generalization of a well–known result due to Newman applicable to an infinite channel. The starting point for the analysis is the very slowly converging Fourier–series expansion of the modified Green' function. This Fourier series can be separated into a Fourier series in one variable that can be summed in closed form, and a double Fourier series that can be partly summed with respect to one index and transformed by application of the Poisson summation formula and integration by residues. The result of applying the Poisson formula can be expressed either as a rapidly converging series of logarithmic functions, or as a rapidly converging Fourier series which can be integrated analytically with respect to the coordinates of the source point. The use of the modified Green' function is demonstrated first by determining potential distributions that arise from specification of uniform current density on two or more of the boundary surfaces. The potential in a cell with two electrodes embedded in opposite sides of the rectangle is then calculated, and circumstances in which this potential approaches the potential in an infinite channel are determined.

For the first time, a theorem on double matrix summability of double conjugate Fourier series is established.

Isometric pictorial representation has long been favored by geologists as a means of displaying three-dimensional surfaces and bodies. A principal advantage over perspective representation is that plotting and measurement can be made readily anywhere on the diagram. Lobeck (1924, p. 120-121) lists the following properties of isometric diagrams. "Measurements may be made to scale anywhere upon the upper surface of the block in the direction of either of the two coordinates, which in the object itself are at right angles to each other. Distances in other directions are not commensurate with each other unless measured along lines parallel to each other, and they are not commensurate with distances measured on the sides of the block. All the lines parallel in the object itself are parallel in the isometric drawing. All lines vertical in the object are vertical in the drawing. If the vertical scale of the drawing is the same as its horizontal scale, then the measurements made in the direction of any of the three coordinates are commensurate. All angles in an isometric drawing are distorted, and even two angles lying in the same plane cannot be compared with each other unless they lie in exactly similar positions. " The I3D subroutines included with this contribution provide an efficient and fast method of displaying a surface—a single valued function of two variables—in isometric or similar pictorial representation. Numerous options are available through the ISPECS and RSPECS parameter arrays. Among the most important options The diagrams can be seen from all theoretically possible viewpoints. The data can be viewed in an unlimited number of rectangular subblocks, all automatically spaced to avoid point overlap (x and y direction spacings are variable). Any rectangular subarea within the data array area may be considered for plotting and any number and configuration of rectangular subblocks within this subarea may be skipped if desired. Scale, line and point spacing, and density of lines and points to be plotted are variable. The vertical (z) scale may be expanded or contracted relative to the horizontal (x, y) scale without restriction. Distances in x and/or y directions can be stretched or foreshortened independently, with scale remaining constant. Surfaces may be plotted with or without the traditional block diagram frame, and major and minor scale ticks are available. A standard labelling routine can be used. Data input is flexible. Points can be arranged in north-south (y direction) or east-west (x direction) grid lines. Possible applications of isometric pictorial representation can be considered within the framework of four general types of surfaces. North-south, east-west, elevation surfaces, such as topographic surfaces, tops and bottoms of beds, buried bedrock surfaces, etc. North-south, east-west, z surfaces, where z is any measurable quantity, such as, for example, thickness or percent of sand in a sedimentary bed, or Bouguer anomalies. X-y-z surfaces, for example equilibrium surfaces in chemical stability diagrams. X-y-f(x, y) surfaces, mathematical functions of two variables, including functions of many terms where the values at different points cannot be readily visualized. Many preceding papers in the Computer Contributions series deal with functions of this type. Examples are the double (two-dimensional) polynomial and Fourier series currently important in trend-surface and harmonic and spectral analysis. Because of wide interest in computer graphics, a number of algorithms have been written which have the capability of generating three-dimensional diagrams (e.g. Rohlf, 1969). Almost all of the algorithms produce perspective rather than isometric diagrams and most have more-or-less serious bugs and severe limitations in point-array size due to rapid increase of execution time with increase in number of points (A.R. Paradis, personal communication). Algorithms to construct isometric diagrams seemingly are less common, as I do not know of any, other than I3D. The I3D subroutines have been subjected to thorough testing, and numerous isometric drawings have been obtained, but it is entirely possible that algorithm defects are present due to the complexity of the subroutines. Acknowledgments Mr. Arthur R. Paradis, in charge of computer graphics at the University of California, Berkeley, Computer Center, gave encouragement and arranged use of computer facilities during the program development. His methodology is evident in both the organization of the latter half of this paper and in the subroutine structuring. The Anaconda Company provided some development computer time and permitted use of the view of the Berkeley open-pit mine in Example 2. Prof. Charles Meyer, Prof. Garniss Curtis, and Mr. Paradis kindly reviewed the manuscript.

ABSTRACTA new benchmark with a high accurate solution is proposed for the verification of numerical codes dealing with double-diffusive convection in a heterogeneous porous medium. The new benchmark is inspired by the popular problem of square porous cavity by assuming a stratified porous medium. A high accurate steady state solution is developed using the Fourier–Galerkin method. To this aim, the unknowns are expanded in double infinite Fourier series. The accuracy of the developed solution is assessed in terms of the truncation orders of the Fourier series. Comparison against finite element solutions highlights the worthiness of the proposed benchmark for numerical code validation.

Leakage inductance is one of the key parameters of a transformer, and it is often intentionally integrated into transformers. Rogowski’s equation is generally used for leakage inductance calculation; however, it is only applicable to concentric winding transformers where windings have the same height. Consequently, it has limited applications. This paper proposes a transformer leakage inductance calculation method using a double Fourier series. The limitation of Rogowski’s leakage inductance equation was analyzed in practical applications, and a new model for calculating the leakage inductance of a double-group-overlapping winding transformer was derived. Experimental prototypes of transformers were developed, and their simulation models were built in Ansys. The correctness of the proposed calculation method for transformer leakage inductance using a double Fourier series was verified by comparing the calculation results with the simulation and measured ones. A sensitivity analysis was also conducted by studying the variations in different parameters that might affect the leakage inductance value. The proposed calculation model gives an intuitive and simple method with less calculation and design effort while maintaining reasonable accuracy for leakage inductance calculation. In addition, the featured double Fourier series approach has a wider range of applications than Rogowski’s equation.

Nonlinear systems can have periodic solutions evolving with the parameters of the system. Studying this evolution (numerical continuation of solutions) uncovers sought-after regimes in musical acoustics : many musical instruments rely on auto-oscillation, that is, the excitation of a nonlinear system coupled with a linear resonator, where some parameters may be adjusted by the player. Periodic solutions can be approximated as truncated Fourier series (Harmonic Balance Method) ; the period is one of the unknowns. Several stable or unstable solutions can be found for the same playing parameters thanks to continuation. An important challenge is the continuation of quasi-periodic solutions, also called multiphonic sounds by musicians. Depending on the context, these oscillation regimes are considered pleasant (jazz or contemporary music for instance) or unpleasant (classical music). We developed a method based on double Fourier series, coupled with a continuation technique. The two base frequencies are unknowns and incommensurable. The system is reformulated as quadratic in order to allow straight interface with previous work on periodic harmonic balance. This method is illustrated on simple models relevant to musical acoustics, though the method can be applied to many nonlinear problems, without a priori knowledge of the solutions.

Lateral distribution of P-wave velocity in Kirishima volcanic region, South Kyushu, is studied assuming a model consisting of a uniform half space and a superficial layer, which is vertically uniform, but horizontally heterogeneous.The reciprocal of velocity in the layer is expressed by a double Fourier series, and the coefficients are determined using the arrival time data observed at the stations belonging to Kirishima Volcano Observatory. Using the coefficients thus obtained P-wave velocity is synthesized, and the result is summarized as follows:1) The velocity of the area lying NNW of Kirishima volcanic zone is low compared with the volcanic area itself.2) Kakuto Basin, situated about 10km NNW of the volcanic area, shows specially low velocity.3) Except in Kakuto Basin the velocity distribution seems not to be related to any of the Bouguer anomaly, surface geology, or topography.4) Equal velocity contours run in NW-SE and NNE-SSW directions, which harmonize with the trends of fault series.The above feature of distribution seems to reflect the inner structure successfully, and also the validity of the method of a double Fourier series for determining the velocity distribution of body waves in the layer.

This publication describes a computer program designed for analysis of surfaces using double Fourier series. Analysis of surfaces in geology is known commonly as trend analysis; in statistics and economics, the technique is regarded as a form of regression analysis. In trend or regression analysis, a mathematical function is fitted to data points so that the function satisfies the least-squares criterion, which is a criterion of "best" fit.

Further study on the high-order double-Fourier-series spectral filtering on a sphere

We introduce semicontinuous summation methods for series of fuzzy numbers and give Tauberian conditions under which summation of a series of fuzzy numbers via generalized Dirichlet series and via generalized factorial series implies its convergence. Besides, we define the concept of level Fourier series of fuzzy valued functions and obtain results concerning the summation of level Fourier series.

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at such a point where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth. We also consider the extension of the Zygmund classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi> λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\Lambda _{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the two-dimensional torus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {T} ^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

We have previously proposed a force-field refinement method. This method consists of minimising the sum of the square of the force acting on each atom in the proteins with the structures from the Protein Data Bank. In addition, we also proposed a new backbone-torsion-energy term, which is represented by a double Fourier series in two variables, the backbone dihedral angles and . In this paper, we apply our optimisation method and backbone-torsion-energy term to AMBER ff94 (ff96) force field for molecular simulation of protein systems. The results imply that the new force-field parameters give structures of two peptides more consistent with the experimental implications for secondary-structure-forming tendencies than the original force field.