About B-splines. Twenty answers to one question: What is the cubic B-spline for the knots -2,-1,0,1,2?

In this composition an attempt is made to answer one simple question only: What is the cubic B-spline for the knots -2,-1,0,1,2? The note will take you on a most interesting trip through various fields of Mathematics and finally convince you on how little we know

On the arclength of trigonometric interpolants

As pointed out recently by Strichartz [5], the arclength of the graph $\Gamma(S_N(f))$ of the partial sums $S_N(f)$ of the Fourier series of a jump function $f$ grows with the order of $\log N$. In this paper we discuss the behaviour of the arclengths of the graphs of trigonometric interpolants to a jump function. Here the boundedness of the arclengths depends essentially on the fact whether the jump discontinuity is at an interpolation point or not. In addition convergence results for the arclengths of interpolants to smoother functions are presented

On the Arclength of Trigonometric Interpolants

As pointed out by Strichartz [5] recently, the arclength of the graph \Gamma(S N (f)) of the partial sums SN (f) of the Fourier series of a jump function f grows with the order of log N . In this paper we discuss the behaviour of the arclengths of the graphs of trigonometric interpolants to a jump function. Here the boundedness of the arclengths depends essentially on the fact whether the jump discontinuity is at an interpolation point or not. In addition convergence results for the arclengths of interpolants to smoother functions are presented. 1 Research of the authors supported by the EU Research Training Network MINGLE, RTN1-199900212 1 1 Introduction The famous Gibbs phenomenon of overshooting is one of the well-known disadvantages of the Fourier series approach. It is closely related to the logarithmic growth of the L 1 2 -norm of the Dirichlet kernel, i.e. the Lebesgue constant for the Fourier partial sum operator. It can also be seen as one of the motivations for introdu..

Applied Wave Mathematics: Selected Topics in Solids, Fluids, and Mathematical Methods

This edited volume addresses the importance of mathematics in wave-related research, and its tutorial style contributions provide educational material for courses or seminars. It presents highlights from research carried out at the Centre for Nonlinear Studies in Tallinn, Estonia, the Centre of Mathematics for Applications in Oslo, Norway, and by visitors from the EU project CENS-CMA. The example applications discussed include wave propagation in inhomogeneous solids, liquid crystals in mesoscopic physics, and long ship waves in shallow water bodies. Other contributions focus on specific math