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The spline theory can be derived from two sources: the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We show that the general variational approach called smooth interpolation introduced by Talmi and Gilat covers not only the cubic spline but also the tension spline (called also spline in tension or spline with tension) in one or more dimensions. We show the results of a 1D numerical example that present the advantages and drawbacks of the tension spline. [ABSTRACT FROM AUTHOR]

We build a numerical approximation method, for Fredholm integral equation solution of the second type. This method is based on the regularization by convolution and Fourier series expansion. It provides a better convergence order. [ABSTRACT FROM AUTHOR]

We propose two methods to find analytic periodic approximations intended for differential equations of Hill type. Here, we apply these methods on the simplest case of the Mathieu equation. The former has been inspired in the harmonic balance method and designed to find, making use on a given algebraic function, analytic approximations for the critical values and their corresponding periodic solutions of the Mathieu differential equation. What is new is that these solutions are valid for all values of the equation parameter q , no matter how large. The second one uses truncations of Fourier series and has connections with the least squares method. [ABSTRACT FROM AUTHOR]

The paper is primarily devoted to the problem of smooth interpolation of data in 1D. In addition to the exact interpolation of data at nodes, we are also concerned with the smoothness of the interpolating curve and its derivatives. The interpolating curve is defined as the solution of a variational problem with constraints. The system of functions exp ( - i kx ) , k being an integer, is taken for the basis of the space where we measure the smoothness of the result. It also generates the functions used for the interpolation itself. Choosing different norms when measuring the smoothness, we arrive at different interpolating functions. We also mention the problem of smooth curve fitting (data smoothing). We discuss the proper choice of different norms for this way of approximation and present the results of several 1D numerical examples that show the advantages and drawbacks of smooth interpolation. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, a new theorem on the degree of approximation of function conjugate to a 2π periodic function f belonging to the generalized weighted Lipschitz -class by dropping the monotonicity condition on the generating sequence {pn } has been established which in turn generalizes the results of Lal (2009) [12] on Fourier series. We also note some errors appearing in the paper of Lal (2009) [12] and rectify them in the view of observations of Rhoades et al. (2011) [22]. [Copyright &y& Elsevier]

Abstract: In order to compute an approximate solution of a weakly singular integral equation, we first regularize the kernel and then truncate the associated Fourier series. Applications to Green and Abel operators are given. [Copyright &y& Elsevier]

Abstract: The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness. [Copyright &y& Elsevier]

Abstract: In a recent paper Lal and Yadov obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using the product of the Cesàro and Euler means of order one of its Fourier series. In this paper we extend this result to any regular Hausdorff matrix for the same class of functions. [Copyright &y& Elsevier]

Abstract: The interior and exterior Neumann problems for the Helmholtz equation in starlike planar domains are addressed by using a suitable Fourier-like technique. Attention is in particular focused on normal-polar domains whose boundaries are defined by the so called “superformula” introduced by Gielis. A dedicated numerical procedure based on a computer algebra system is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson. [Copyright &y& Elsevier]

Abstract: In this paper, two new theorems on the degree of approximation of the function f ∈Lipα and f ∈ W(L r , ξ(t)) have been established. [Copyright &y& Elsevier]

Abstract: In this study, by obtaining some Korovkin type approximation results in statistical sense for certain positive linear operators constructed by means of the Chan–Chyan–Srivastava multivariable polynomials [W.-C.C. Chan, C.-J. Chyan, H.M. Srivastava, The Lagrange polynomials in several variables, Integral Transform. Spec. Funct. 12 (2001) 139–148], we show that our approximation method is stronger than the corresponding classical aspects in the approximation theory settings. Furthermore, we investigate their statistical rates by means of the modulus of continuity and the elements of the Lipschitz class. [Copyright &y& Elsevier]

Recently, L.N. Mishra et al. obtained a new theorem on the degree of approximation of function f ̃ : conjugate to a 2 π periodic function f belonging to the generalized weighted Lipschitz W ( L r , ξ ( t ) ) ( r ⩾ 1 ) -class (Mishra et al., 2014). In this note, we point out that the conclusion of the theorem only holds for the constant functions, and thus the result is trivial. [ABSTRACT FROM AUTHOR]