Equidistant Knots Research Articles

Error Bounds for Gauss Type Quadrature Formulas Related to Spaces of Splines with Equidistant Knots

Error bounds for the Gauss type quadrature formulae QGn, QLn+1 and QRn+1 (Gauss, Lobatto and Radau formulae) related to the spaces of polynomial spline functions of degree r−1 with equidistant knots are obtained. It is shown that these quadrature rules are asymptotically optimal in the Sobolev space Wr∞ for all r, and in Wrp (1 ≤ p ≤ ∞) for odd r. Some inequalities involving the Gaussian nodes and weights are also established.

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL2[−1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.

BestL 1-approximations of classesW 1 r by Splines fromW 1 r

We obtain the exact values of the bestL1-approximations of the classesW 1 r of periodic functions by periodic polynomial splines of degreer and defect 1 with equidistant knots that belong to the classW 1 r .

Gauss Quadratures: An Inverse Problem

The problem of finding the Gauss quadrature (i.e., the quadrature formula of the maximal polynomial order) for a given weight function is reversed: a weight function, for which a given quadrature formula with positive weights is the Gauss quadrature, is sought. Among all such weight functions, those minimizing, in the least square sense, the kth derivative ($k > 0$ given) are characterized. An algorithm for calculating the values of the minimizing weight functions is derived and applied to find positive weights for quadratures with equidistant knots. The concepts are further generalized to include Gauss–Radau and Gauss–Lobatto quadratures.

A connection between quartic spline solution and numerov solution of a boundary value problem

This brief paper derives via quartic spline function a new consistency recurrence relation connecting the quartic spline function values at equidistant knots and the corresponding values of the second derivatives. It is shown how this consistency relation may be used in an algorithm for computing quartic spline approximations to the solution and its higher derivatives for a linear two-point boundary value problem. Some numerical evidence is also included to demonstrate the practical usefulness of the algorithm.

Zur konstruktion periodischer hermite-interpolationssplines bei äquidistanter knotenverteilung

A complex line integral representation of the fundamental splines is used as a starting point for a simple and lucid method for the construction of periodic interpolating Hermite-splines with equidistant knots. Several applications and generalizations are indicated.

Condition of boundary integral equations arising from flow computations

Stating flow boundary conditions for normal and tangential velocities leads to Fredholm integral equations of the first and second kind. Approximating the vortex distribution on pressure and suction side by piecewise linear functions with equidistant knots, collocation yields a commonly overdetermined system of linear equations with the vortex densities as unknowns. There are several possibilities for collocation. The question arises what kind of integral equation and what collocation points are to be preferred. Our main conclusions based on condition numbers of the resulting linear systems are: - The equation of the first kind with Cauchy singular kernel does not appear to be worse conditioned than the equation of the second kind with regular kernel. - Collocation points for the equation of the first kind are better placed in between the knots, for the equation of the second kind at the knots. - The circulation is mainly determined by the equation of the first kind, the mean velocity level by the equation of the second kind. Employing simultaneously equations of the first and second kind generally improves the solution and allows to cope with aerodynamic profiles of any thickness and loading.

Quadratic Spline and Two-Point Boundary Value Problem

where f ( t , x, y) is defined and twice continuously differentiable in a region D of the (t, x} :y)-space intercepted by two hyperplanes £=0 and t=l. The analytical solution of (1.1) for arbitrary choices of the function / cannot be found in general. We usually resort to some numerical method for obtaining an approximate solution of the problem (1.1). The standard numerical methods for the numerical treatment of (1.1) consist of finite difference methods, shooting methods, RayleighRitz and Galerkin's procedure and collocation methods. A long list of references of all of these methods is given by Keller [4] in [2]. The subject of obtaining spline solutions for the initial as well as boundary value problems is briefly discussed in [1]. Since then many papers have appeared dealing with the continuous approximation of x(t) satisfying (1.1) via cubic and quintic splines mainly (see [3, 9]). The collocation methods using spline functions have been developed and analysed by Sakai (see [5], [6], [7]) again employing cubic and quintic splines at equi-distant knots. Recently Usmani and Sakai [11] have also used quadratic spline function for solving a two-point boundary value problem involving a fourth order differential equation. In this brief report, we propose a second order collocation method using quadratic spline employing 5-splines. Ill the sequel, it will be shown that our method is an 0(/i)-convergent procedure. In the last section some numerical evidence is included to show the practical applicability of our method by solving

Gruppentheoretische aspekte der signalübertragung und der kardinalen interpolationssplines I

AbstractAs is well known the real Heisenberg nilpotent group A(R) constitutes the group‐theoretic embodiment of the Heisenberg canonical commutation relations (CCR) of classical quantum mechanics. In this connection, quantum mechanics stands for the quantum‐mechanical description, at a given instant of time, of a non‐relativistic microparticle moving in the one‐dimensional configuration space R and having the plane R2 as its (flat) phase space. In fact, the (nilpotent) Lie algebra n of Ã(R) reflects the Weyl equations which are global versions of the Heisenberg CCR. Unfortunately, the subject of Heisenberg nilpotent groups is outside quantum mechanics and all the more outside mathematical physics not as commonly known as it should be considering its wide range of applications in a variety of different fields. The present paper which has two parts aims to develop a central topic of nilpotent harmonic analysis, to wit, the microparticle model, the lattice model which will be realized on the Heisenberg compact nilmanifold, and the complex wave model (or Bargmann‐Fock‐Segal model) of the linear Schrödinger representation of Ã(R) in order to examine geometrically several applications which are governed by the real Heisenberg nilpotent group Ã(R). These applications are in Part I the classical Whittaker‐Shannon sampling theorem which is of basic importance in signal processing, to wit, for the transmission of digital signals as well as analog signals, and in Part II the Subbotin‐Schoenberg existence and uniqueness theorem of cardinal spline interpolation. Moreover, Part II indicates briefly some connections of the aforementioned models to the Wigner phase‐space quasiprobability density function of quantum statistical mechanics via the Schwartz kernels theory on unimodular Lie groups, an approach to the cross‐ and autoambiguity functions of radar synthesis, and to the Zak transform of solid state physics. The second part also points out some relations of harmonic analysis of the finite nilpotent group A(Z/NZ) to periodic spline interpolants admitting N equidistant knots on the one‐dimensional compact torus group T. These last examples should serve mainly as hints for some further lines of investigations in the field of applications of nilpotent harmonic analysis.

Discrete cubic spline interpolation

In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.

Moments and Fourier transforms of B-splines

The problem of determining the moments and the Fourier transforms of B-splines with arbitrary knots is considered. There exists a simple connection between the moments of such splines and the so-called extended Stirling numbers of the second kind which are defined in section 2. Some recurrence relations for the moments of B-splines with arbitrary knots are given in section 3. In the case of equidistant knots we have also further recurrences. For the forward, central and perfect B-splines the explicit formulas for the moments are given in section 3. The Fourier transforms of B-splines is treated in section 4. The final section is devoted to so-called Stieltjes series connected with the nonnegative weight function w(x) and such that ∫ a b w(x) dx > 0 in some closed interval [a, b]. It is proved that such series for the particular values of the independent variable may be expressed by the finite sums which contain the nodes and coefficients of the optimal (in the Davies sense) quadrature formulas.

On deficient cubic spline interpolation

Following Schoenberg [5] and de Boor [2] one of the present authors [3] has treated cubic spline interpolation by matching the integral mean of a function and a spline between successive equidistant knots. Earlier Sharma and Tzimbalario [6] had studied quadratic splines with similar matching conditions. Our object is to study deficient cubic splines by making less restrictive continuity requirements at the joints and having two interpolatory conditions, one of which is the matching condition at appropriate points of the dividing intervals while the other is the matching of the integral means. Corresponding error bound is also obtained in the present paper. It may be mentioned here that sharp error bounds in different norms have been given by Varga in [7; Theorem 2.11, where spline interpolation of bounded linear functionals is studied.

Upper bounds of best one-sided approximations of the classes WrLΨ in the metric of L

The lowest upper bound is obtained for best one-sided approximations of classes (r=1,2 ...) by trigonometric polynomials and splines of minimum deficiency with equidistant knots, in the metric of space L, where WrLΨ={f:f(x+2π)=f(x), f(r−1)(x) is absolutely continuous, ∥f(r)∥LΨ⩽1} and LΨ is an Orlicz space.

The L1 norm of the approximation error for splines with equidistant knots

This paper is concerned with the estimation of the L 1 norm of the difference between a function f of bounded variation in [0, 1] and the associated variation-diminishing spline function with equidistant knots, S m,n f, with see Schoenberg [4]. For f bounded on [0, 1] and for integers m, n such that $$ n \geqslant m \geqslant 2, $$ (1.1) the function S m,n f is defined by $$ S_{m.n} f\left( x \right) = \sum\limits_{j = 0}^\iota {f\left( {n^{ - 1} \xi \left( {m,j} \right)} \right)\bar N_{m.j} \left( {nx} \right),} $$ (1.2) where $$ l = m + n - 2, $$ (1.3) $$\xi \left( {m,j} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\left( {j + 1} \right)j}}{{2\left( {m - 1} \right)}},} \hfill & {j = 0, \ldots ,m - 2} \hfill {j + 1 - \frac{m}{2},} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill {n - \xi \left( {m,l - j} \right),} \hfill & {j = n, \ldots ,l,} \hfill \end{array} } \right.$$ (1.4) $${{\tilde{N}}_{{m,j\left( x \right)}}} = \left\{ {\begin{array}{*{20}{c}} {\frac{{j + 1}}{m}{{h}_{{m.j + 1}}}\left( x \right),} \hfill & {j = 0, \ldots ,m - 2,} \hfill {{{h}_{m}}\left( {x + m - j - 1} \right),} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill {{{{\bar{N}}}_{{m.l - j}}}\left( {n - x} \right),} \hfill & {j = n, \ldots ,l,} \hfill \end{array} } \right.$$ (1.5) $${{h}_{{m,k}}}\left( x \right) = \frac{m}{{k!}}\sum\limits_{{i = 1}}^{k} {{{{( - 1)}}^{{k - i}}}{{i}^{{k - m}}}} \left( {_{i}^{k}} \right)\left( {i - x} \right)_{ + }^{{m - 1}}{\mkern 1mu} {\text{for }}x0,$$ (1.6) $$ h_m \left( x \right) = h_{m.m} \left( x \right).$$ (1.7) .

Explicit calculation of interpolating cubic splines on equi-distant knots

The coefficients of the cubic splines(x) interpolating to the functionf(x) on the equi-distant knots,xi=ih(i=0(1)n andh=1/n) in the interval [0, 1], are determined explicitly in the cases whenf(x) is either periodic or has linear combinations of the first and second derivatives specified as boundary conditions.

On the Convergence of Trigonometrical Interpolation at Equi-Distant Knots

where f(x) is the function interpolated. Naturally the question arises, under what conditions the polynomial T5(x) converges to the function f(x) when n is increasing. Papers dealing with this problem generally make some assumptions concerning the nature of the function f(x) in the whole segment [0, 27r]. Such a treatment makes it possible to apply the results established in the theory of the best approximation of functions and leads to different tests of uniform convergence of T0 (x) towards f(x). These tests prove very similar to those of uniform convergence of Fourier-series. Yet there is also a second possibility of treatment which it is natural to call a local one. Namely, we may examine properties of f(x) near a fixed point X0 which will cause the convergence of Tn(xo) towards f(xo). I know of only one paper treating the problem in this way, namely the classic memoir by Vallee-Poussin.' Vallee-Poussin has proved that the equality