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  • 1
    Electronic Resource
    Electronic Resource
    Polynomial interpolation in several variables (2000)
    Springer
    Advances in computational mathematics 12 (2000), S. 377-410 
    Details
    ISSN: 1572-9044
    Keywords: interpolation ; multivariate polynomials ; Newton approach ; divided differences ; Gröbner bases ; H-bases
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is a survey of the main results on multivariate polynomial interpolation in the last twenty-five years, a period of time when the subject experienced its most rapid development. The problem is considered from two different points of view: the construction of data points which allow unique interpolation for given interpolation spaces as well as the converse. In addition, one section is devoted to error formulas and another to connections with computer algebra. An extensive list of references is also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018981505752
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  • 2
    Electronic Resource
    Electronic Resource
    H-bases for polynomial interpolation and system solving (2000)
    Springer
    Advances in computational mathematics 12 (2000), S. 335-362 
    Details
    ISSN: 1572-9044
    Keywords: ideal bases ; Gröbner bases ; multivariate polynomials ; interpolation ; systems of polynomial equations ; 65D05 ; 65H10 ; 13P10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The H-basis concept allows, similarly to the Gröbner basis concept, a reformulation of nonlinear problems in terms of linear algebra. We exhibit parallels of the two concepts, show properties of H-bases, discuss their construction and uniqueness questions, and prove that n polynomials in n variables are, under mild conditions, already H-bases. We apply H-bases to the solution of polynomial systems by the eigenmethod and to multivariate interpolation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018937723499
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  • 3
    Electronic Resource
    Electronic Resource
    Squaring the Triangle (2000)
    Springer
    Geometriae dedicata 83 (2000), S. 229-243 
    Details
    ISSN: 1572-9168
    Keywords: generalized quadrangle ; projective plane ; circle plane ; interpolation ; unisolvent sets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We construct flat Laguerre planes by ‘integrating’ flat projective planes. The construction is based in an essential way on results from the theory of interpolation. In conjunction with the unifying theory of topological circle planes and generalized quadrangles, the new construction appears to be one of the most natural and powerful constructions of such geometries.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005296310226
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  • 4
    Electronic Resource
    Electronic Resource
    Cesaro Summability with Respect to Two-Parameter Walsh Systems (2000)
    Springer
    Monatshefte für Mathematik 131 (2000), S. 321-334 
    Details
    ISSN: 1436-5081
    Keywords: 2000 Mathematics Subject Classifications: 42C10 ; 43A75 ; 42B08 ; 42B30 ; Key words: Walsh functions ; Hardy spaces ; (C ;  1) summation ; p-atoms ; interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract.  The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H p to L p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L p spaces to L p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s006050070004
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  • 5
    Electronic Resource
    Electronic Resource
    Spatial shape‐preserving interpolation using ν‐splines (2000)
    Springer
    Numerical algorithms 23 (2000), S. 217-250 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; shape‐preserving ; splines ; ν‐spline ; space curves ; 65D05 ; 65D07 ; 65D17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a global iterative algorithm for constructing spatial G 2‐continuous interpolating ν‐splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of ν‐splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019156202082
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  • 6
    Electronic Resource
    Electronic Resource
    Rational interpolation through the optimal attachment of poles to the interpolating polynomial (2000)
    Springer
    Numerical algorithms 23 (2000), S. 315-328 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; optimal interpolation ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019168504808
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  • 7
    Electronic Resource
    Electronic Resource
    A matrix for determining lower complexity barycentric representations of rational interpolants (2000)
    Springer
    Numerical algorithms 24 (2000), S. 17-29 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; barycentric representation ; barycentric weights ; complexity ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019180807534
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  • 8
    Electronic Resource
    Electronic Resource
    Newton-Thiele's rational interpolants (2000)
    Springer
    Numerical algorithms 24 (2000), S. 141-157 
    Details
    ISSN: 1572-9265
    Keywords: continued fraction ; interpolation ; algorithm ; 41A20 ; 65D05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is well known that Newton's interpolation polynomial is based on divided differences which produce useful intermediate results and allow one to compute the polynomial recursively. Thiele's interpolating continued fraction is aimed at building a rational function which interpolates the given support points. It is interesting to notice that Newton's interpolation polynomials and Thiele's interpolating continued fractions can be incorporated in tensor‐product‐like manner to yield four kinds of bivariate interpolation schemes. Among them are classical bivariate Newton's interpolation polynomials which are purely linear interpolants, branched continued fractions which are purely nonlinear interpolants and have been studied by Chaffy, Cuyt and Verdonk, Kuchminska, Siemaszko and many other authors, and Thiele-Newton's bivariate interpolating continued fractions which are investigated in another paper by one of the authors. In this paper, emphasis is put on the study of Newton-Thiele's bivariate rational interpolants. By introducing so‐called blending differences which look partially like divided differences and partially like inverse differences, we give a recursive algorithm accompanied with a numerical example. Moreover, we bring out the error estimation and discuss the limiting case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019193210259
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  • 9
    Electronic Resource
    Electronic Resource
    Interpolation by Rational Functions with Nodes on the Unit Circle (2000)
    Springer
    Acta applicandae mathematicae 61 (2000), S. 101-118 
    Details
    ISSN: 1572-9036
    Keywords: orthogonal rational functions ; interpolation ; R-Szegő quadrature
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract From the Erdős–Turán theorem, it is known that if f is a continuous function on $$ {\Bbb T} = \left\{ {z:\left\lfloor z \right\rfloor = 1} \right\} $$ and L n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then $$ \mathop {\lim }\limits_{n \to \infty } \int_{\Bbb T} {\left| {f\left( z \right)} \right|^2 } \left| {{\text{d}}z} \right| = 0 $$ Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in $$ {\Bbb D} = \left\{ {z:\left| z \right| 〈 1} \right\} $$ and continuous on $$ {\Bbb D} \cup {\Bbb T} $$ and making use of algebraic interpolating polynomials in the roots of unity. In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on $$ {\Bbb T} $$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1006433627981
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