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  • 1
    Electronic Resource
    Electronic Resource
    On the convergence of interpolating periodic spline functions of high degree (1972)
    Springer
    Numerische Mathematik 19 (1972), S. 146-154 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let the spline functionS m of degree 2m−1 and period 1 be the unique solution of the interpolation problem in § 1. An interesting question was posed by Schoenberg [1], p. 125: What happens toS m if we letm→∞? In this paper, we prove that the spline functionsS m and their derivatives converge form→∞ to a well determined trigonometric polynomial and its derivatives. Estimates for the rate of convergence are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01402525
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  • 2
    Electronic Resource
    Electronic Resource
    Applications of shortest path algorithms to matrix scalings (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 111-126 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F35 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A symmetric scaling of a nonnegative, square matrixA is a matrixXAX −1, whereX is a nonsingular, nonnegative diagonal matrix. By associating a family of weighted directed graphs with the matrixA we are able to adapt the shortest path algorithms to compute an optimal scaling ofA, where we call a symmetric scalingA′ ofA optimal if it minimizes the maximum of the ratio of non-zero elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389759
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  • 3
    Electronic Resource
    Electronic Resource
    Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 65-84 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.13
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont≡1 then the minimum of Problem A is computed by equality (2.4). Ift≡1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont≡1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift≢1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset≡1 but unknown in caset≢1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399312
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  • 4
    Electronic Resource
    Electronic Resource
    Trigonometric polynomials with many real zeros (1985)
    Springer
    Constructive approximation 1 (1985), S. 1-13 
    Details
    ISSN: 1432-0940
    Keywords: 42A05 ; 42A10 ; 26C05 ; Trigonometric polynomials ; Inequalities ; Zeros ; Geometric convergence ; Bounded analytic functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This is a contribution to the theory of “incomplete trigonometric polynomials”T n , but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some intervalI whose length is not too large. We begin with the simple theorem that if ∥T n ∥ ≤ 1 and ifT n has ≥θn, 0〈θ〈 2, zeros at 0, thenT n (t) must be small on the interval |t|〈2 arcsin (θ/2). There are similar but more complicated and more difficult to prove results whenT n has ≥θn zeros onI. These results have the following application: IfT n →f a.e., and if ∥T n 〉∥∞〈-1, thenf vanishes on a set of the circleT whose measure is controlled by lim sup (N n /n), whereN n is the number of zeros ofT n onT. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01890020
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    Paper (German National Licenses)
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