Concise Numerical Mathematics

A non-iterative approach to frequency estimation of a complex exponential in noise by interpolation of fourier coefficients

“…For this, first the residual being calculated in step one is removed, so in order to run the algorithm, = 0 is taken as a fixed point. In this step algorithm converges to a fixed point following the fixed point theorem ( [15], pp.109-111). However as stated earlier the algorithm gives the worst estimation in terms of variance at this point.…”

Superconvergence in collocation methods for Volterra integral equations with vanishing delays

“…It should be pointed out that for arbitrary collocation parameters {c i }, the precision of the interpolatory m-point quadrature formula over [0, 1] is m − 1 generally. But if the orthogonality condition (3.20) holds, then the corresponding precision will attain m at least (see [20]). Using the assumed regularity of the given functions, the orthogonality condition of the collocation points, and the fact that For the second term I 2 , we set…”

The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind

“…The standard error representation for polynomial interpolation (see e.g., [18]) gives |P(0) − ψ(0)| ≤ max 0≤x≤2h |ψ (x)|h 2 . In addition, an expansion of P ε (0) − P(0) in terms of Lagrange basis polynomials L n ∈ 1 , n = 1, 2, with respect to the two grid points h, 2h gives |P ε (0) − P(0)| ≤ cε with c = |L 1 (0)| + |L 2 (0)| = 3.…”