On determination of jumps in terms of Abel-Poisson mean of Fourier series

In this paper, we generalize some well-known results (Theorems A, C, and D) by establishing two general results (Theorems 1 and 3). As special applications, we find that the (generalized) jumps of f can be determined by the higher order partial derivatives of its Abel-Poisson means. This is different from the determination of jumps by higher order derivatives of the partial sums. We also give some estimates of the higher order partial derivatives of the Abel-Poisson mean of an integrable function F at those points at which F is smooth.     

More on determination of jumps

We generalize some old and new results on the determination of jumps of a periodic, Lebesgue integrable function f at each point of discontinuity of first kind in terms of the partial sums of the conjugate series to the Fourier series of f in [1], and in terms of the Abel-Poisson means in [2], to some more general linear operators which satisfy some certain conditions. The linear operators in discussion include the Fejér means, de la Vallée-Poussin means, and Bernstein-Rogosinski sums. Research also supported in part by NSF of China under grant number 10471130.     

The absolute convergence of the Fourier-Haar series for two-dimensional functions

It is well-known that if an one-dimensional function is continuously differentiable on [0, 1], then its Fourier-Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives f _x ^′ and f _y ^′ on T ², then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier-Haar series for two-dimensional continuously differentiable functions.     

Symmetric marching technique and elliptic equations with variable coefficient involving mixed partial deriavatives

The symmetric marching technique developed in [1,2] has been extended to the elliptic equations with variable coefficients involving mixed partial derivatives. The restricted class of such equations of the form u_xx+2B(x, y)u_xy+c(x,y)u_yy=0 have been considered. Numerical results of model problems solved are presented.     

Fourier series of half-range functions by smooth extension

This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L] of the interval [−L, L] in one-dimensional problems and over the sub-domain [0, L_x] × [0, L_y] of the domain [−L_x, L_x] × [−L_y, L_y] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.     

On the convergence of double Fourier series of functions in L P[0, 1]2, where p=(p 1, p 2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L _p[0,2]²was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L _p[0,1]², p=(p₁,p₂), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.     

On Fourier series for higher order (partial) derivatives of functions

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

Approximation by Nörlund means of double Fourier series for Lipschitz functions

We study the rate of uniform approximation by Nörlund means of the rectangular partial sums of the double Fourier series of a function |(x, y) belonging to the class Lip α, 0 < α 1, on the two-dimensional torus −π < x, y π. As a special case we obtain the rate of uniform approximation by double Cesàro means.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-convergence of complex double fourier series

It is proved that the complex double Fourier series of an integrable functionf(x, y) with coefficients c_jk satisfying certain conditions, will converge in L¹-norm. The conditions used here are the combinations of Tauberian condition of Hardy-Karamata kind and its limiting case. This paper extends the result of Bray [1] to complex double Fourier series. An erratum to this article is available at .     

Absolutely convergent Fourier series and function classes

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0<α?1, or to one of the Zygmund classes Λ_∗(1) and λ_∗(1). Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f.     

One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10

A one-step 9-stage Hermite–Birkhoff–Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order differential equations of the form y′=f(x,y), y(x ₀)=y ₀. The method uses y′ and higher derivatives y ⁽²⁾ to y ⁽⁴⁾ as in Taylor methods and is combined with a 9-stage Runge–Kutta method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to Taylor- and Runge–Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than Dormand–Prince DP(8,7)13M. The stepsize is controlled by means of y ⁽²⁾ and y ⁽⁴⁾. HBT(10)9 is superior to DP(8,7)13M and Taylor method of order 10 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. These numerical results show the benefits of adding high-order derivatives to Runge–Kutta methods.     

A one-step 7-stage Hermite-Birkhoff-Taylor ODE solver of order 11

A one-step 7-stage Hermite-Birkhoff-Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y^′=f(t,y), y(t₀)=y₀. The method adds the derivatives y^′ to y⁽⁶⁾, used in Taylor methods, to a 7-stage Runge-Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge-Kutta-type order conditions. These conditions are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than the Dormand-Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge-Kutta methods.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (?_yyu₁,?_xxu₂) and the partial magnetic diffusion (?_yyb₁,?_xxb₂). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.     

A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems

Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h² uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process. We present a new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems: y″ = f(x, y, y′), a < x < b, α₀y(a) − α₀y′(a) = A, β₀y(b) + β₁y′(b) = B. We generate the solution at the nodal points by a fourth-order method and then use ‘conditions of continuity’ to obtain smoothed approximations for the second derivatives of the solution needed for the construction of the cubic spline solution. We show that our method provides order h⁴ uniformly convergent approximations over [a, b]. The fourth order of the presented method is demonstrated computationally by two examples.     

Fejér type theorems for Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2], then the nth partial sum of its formally differentiated Fourier series divided by n converges to ^-1 [F(x+0) - F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of nonperiodic functions of bounded variation is also known. These theorems can be interpreted in such a way that the terms of the Fourier-Stieltjes (or Fourier) series of F determine the atoms of the finite Borel measure on the torus T:= [0, 2) induced by an appropriate extension of F (or by F itself in the periodic case). The aim of the present paper is to extend all of these results to the Cesàro as well as Abel-Poisson means of Fourier-Stieltjes (or Fourier) series of a nonperiodic (or periodic) function F of bounded variation. At the end, we sketch a possible extension of these results to linear means defined by more general kernels.     

The extended Bitsadze-Lavrent’ev-Tricomi boundary value problem

F. G. Tricomi ([5], [6]) originated the theory of boundary of value problems for mixed type equations by establishing the first mixed type equation known asthe Tricomi equation \(y \cdot u_{xx} + u_{yy} = 0\) which is hyperbolic fory<0, elliptic fory>0, and parabolic fory=0 and then observed that this equation could be applied in Aerodynamics and in general in Fluid Dynamics (transonic flows). See: M. Cribario [1], G. Fichera [2], and our doctoral dissertation [4]. Then M. A. Lavrent’ev and A. V. Bitsadze [3] established together a new mixed type boundary value problem for the equation \(\operatorname{sgn} (y) \cdot u_{xx} + u_{yy} = 0\) where sgn (y)=1 fory>0, =?1 fory<0, fory=0, which involved thediscontinuous coefficient K=sgn (y) ofu _xx while in the case of Tricomi equation the corresponding coefficientT=y wascontinuous. In this paper we establish another mixed type boundary value problem forthe extended Bitsadze-Lavrent’ev-Tricomi equation \(L u = \operatorname{sgn} (y) \cdot u_{xx} + \operatorname{sgn} (x) \cdot u_{yy} + r (x,y) \cdot u = f (x,y)\) where both coefficientsK=sgn (y),M=sgn (x) ofu _xx ,u _yy , respectively are discontinous,r=r (x, y) is once continuously differentiable,f=f (x, y) continuous, and then we prove a uniqueness theorem for quasi-regular solutions.     

Properties of Cesàro means of double Fourier series

The pointwise behavior of partial sums and Cesàro means of trigonometric series were studied in many papers. This paper deals with the behavior of rectangular Cesàro means at a point (x ₀, y ₀) for functions f(x, y) bounded in the square [?π; π]² and satisfying the condition |f(x ₀ + s, y ₀ + t) ? f(x ₀, y ₀)| ≤ ρ $ \left( {\sqrt {s^2 + t^2 } } \right) $ ^α , for some α ∈ (0, 1) and all s and t.