Generalized Spline Functions Research Articles

Numerical Approach of Linear Volterra Integro-Differential Equations Using Generalized Spline Functions

This paper is dealing with non-polynomial spline functions "generalized spline" to find the approximate solution of linear Volterra integro-differential equations of the second kind and extension of this work to solve system of linear Volterra integro-differential equations. The performance of generalized spline functions are illustrated in test examples

Numerical Approach of Linear Volterra Integro-Differential Equations Using Generalized Spline Functions

This paper is dealing with non-polynomial spline functions "generalized spline" to find the approximate solution of linear Volterra integro-differential equations of the second kind and extension of this work to solve system of linear Volterra integro-differential equations. The performance of generalized spline functions are illustrated in test examples

A remarkable equality referring to spline functions in Hilbert spaces

In the introduction of this paper is presented the definition of the generalized spline functions as solutions of a variational problem and are shown some theorems regarding to the existence and uniqueness. The main result of this article consists in a remarkable equality verified by the generalized spline elements, based on the properties of the spaces, operator and interpolatory set involved, which can be used as a characterization theorem of the generalized spline functions in Hilbert spaces.

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.

Interpolation by Uni- and Multivariate Generalized Splines

Lagrange interpolation by finite-dimensional spaces of uni- and multivariate generalized spline functions (including polynomial splines) is studied. Using a condition of Schoenberg-Whitney type, it is shown how to change an almost interpolation set in order to obtain a set which admits unique Lagrange interpolation. Moreover, it is shown that every regular space of univariate generalized splines is a weak Chebyshev space if and only if every interpolation set can be characterized by a modified Schoenberg-Whitney type condition.

Generalized function estimation of underwater transient signals

In this paper, a generalized approach to the transient signal characterization problems will be presented. Specifically, the class of transient signals as a subspace of an appropriate Hilbert space, namely the Sobolev space, which consists of time-limited and bandlimited signals will be characterized. It is shown that, under this setting, the optimal representation for the general case of transient signals both in the time and frequency domains is the class of generalized spline functions. In addition, to provide some motivational tool for studying the transient signal characterization problem, one specific class of transient signals, which is emitted from submarines, will be focused upon.

Principal components analysis of sampled functions

This paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values. This problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time. In such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful, as for example in the analysis of growth or learning curves. It is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis. The reproducing kernel for the Hilbert space of functions plays a central role, and defines the best interpolating functions, which are generalized spline functions. An example is offered of how sensitivity to derivative information can reveal interesting aspects of the data.

Weak Chebyshev spaces and best L1-approximation

In the first part of this paper it is shown that a large class of weak Chebyshev subspaces of C¦a, b¦ can be considered as a class of generalized spline functions. This generalizes a result of Bartelt ( J. Approx. Theory 14 (1975), 30–37). In the second part L 1-uniqueness for a large subclass of generalized spline functions is proved. In particular, results of Galkin ( Math. Notes 15 (1974), 3–8). Strauss (“Numerische Methoden der Approximationtheorie,” Band 2. 1973), and Carroll Braess ( J. Approx. Theory 12 (1974), 362–364) who have shown L 1, uniqueness for subspaces of polynomial spline functions and certain continuously composed Chebyshev subspaces are obtained in this way.

Closed images of convex multivalued mappings in linear topological spaces with applications

A closedness criterion for the image of a convex closed locally compact set under a convex multivalued mapping is proved. Applications are given to the solvability of linear systems over cones, the existence of generalized spline functions, and the duality theory of abstract mathematical programming.

Asymptotically optimal approximation in fractional Sobolev spaces and the numerical solution of differential equations

We study two extremal problems in fractional Sobolev spaces which were motivated by the consideration of optimal finite difference methods for approximating the solution of a well posed Cauchy problem for linear partial differential equations. The limiting forms as the mesh increment tends to zero of the solutions of the extremal problem are shown to be generalized spline functions. Computations illustrating the effectiveness of these generalized spline functions are included.

On uniform approximation by certain generalized spline functions

On uniform approximation by certain generalized spline functions

Optimal spline digital simulators of analog filters

A now approach is presented for obtaining a linear timevarying recursive digital filter which will optimally simulate the behavior of a linear time-varying analog filter. The property of the best approximation of linear functionals by means of generalized spline functions is invoked in the derivation of the results. In this approach, the optimal digital simulator is, viewed as a min-max estimator of the samples of the analog filter output based on the samples of its input. These samples are not required to appear at uniformly spaced instants of time. The assumption is made that the analog filter input belongs to a class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> of signals, each member of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Omega_{a}</tex> being generated by a known differential dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> forced by some input whose energy is less than or equal to a constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha^{2}</tex> . By interpolating the samples of the analog filter input by a generalized spline associated with the differential operator pertaining to the system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Lambda</tex> , the structure of the optimal digital simulator is derived. Error bounds are derived as functions of the maximum sampling subinterval length and the theory is illustrated by means of examples.