where for n 1, 2 . . . . we have (1 .2) 1 fix,,,>xz„> . . .>x,,,, Then, as it is well known, for given values y,,,, there is exactly one polynomial g(x) of degree ~ aI such that g(x, •„)(r'1, 2, . . ., n) . If the values y, ., . are the values f(x,,,,) of a function f(x) defined in [-1, 1], then we call the corresponding g(x) polynomial "the nr'l interpolatory polynomial of f(x) belonging to A"...

such an approach is local. We can choose the interpolation points nonuniformly by assigning more interpolation points around irregular locations. The disadvantage of ISS is that the generated curves are not polynomial splines. V. CONCLUSION In this correspondence, we propose the ISS approach for the initialization of a wavelet transform, which is very important for further applications. Moreove...

Detecting all local extrema or the global extremum of a polynomial on the torus, the sphere or the rotation group is a tough yet often requested numerical problem. We present a heuristic approach that applies common descent methods like nonlinear conjugated gradients or Newtons methods simultaneously to a large number of starting points. The corner stone of our approach are FFT like algorithms,...

We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct ...

Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms and, thereby, associated scales of interpolatory inequalities. Using one parameter families of index functions based on the dilations of given index functions, new classes of interpolatory inequalities, dilational interpolatory inequalities (DII), are constructed. They have ord...

This paper provides several constructions of compactly supported wavelets generated by interpolatory reenable functions. It was shown in D1] that there is no real compactly supported orthonormal symmetric dyadic reenable function, except the trivial case; and also shown in L] and GM] that there is no compactly supported interpolatory orthonormal dyadic reenable function. Hence, for the dyadic d...

The objective of this paper is to study and construct matrix-valued templates for interpolatory curve subdivision. Since our investigation of this problem was motivated by the need of such subdivision stencils as boundary templates for interpolatory surface subdivision, we provide both spline and non-spline templates that are necessarily symmetric, due to the lack of direction-orientation in ca...

We apply polynomial approximation methods—known in the numerical PDEs context as spectral methods—to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right-hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residual-minimizing Galerkin method, and we show how each can be interpreted as solving a...

We propose a general study of the convergence of a Hermite subdivision scheme H of degree d > 0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme S. The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of S is contractive, then S is C...