Sobolev norms

It is difficult to estimate the approximation error between f and f analytically. The primary approach to minimise this error is to minimise the error caused by the interpolation. Mallat and Zhong selected the interpolation function used in Step 3, based on the minimisation of the interpolation error in Sobolev norm. In the numerical computations presented in this chapter, the same interpolation scheme was used. 

A key foundation of the multi-scale measures is the homogeneous Sobolev norms [9,10]... 

That is, these norms can be considered as weighted sums of the coefficients in the Fourier decomposition of an instantaneous field /(x). While the most straightforward application is thus to spatially periodic domains, it has been shown [10] that for = — 1, 0, 1 these homogeneous Sobolev norms can be applied to arbitrary (nonperiodic) domains. 

All negative Sobolev norms display behavior similar to Os [10], and thus all measures of the form... 

An explicit convergence proof for Sobolev norms can be found in [15). Let us fix the sampling nodes and increase the domain fi. What we observe is that, similar to the surface spline, the bicubic B-spline interpolant tends to a linear polynomial. 

It has been shown [14] for both types ofbasis sets (1.1) and (1.2) that a given set of dimension n can be regarded as a member B of a family of basis sets that in the limit n oo become complete both in the ordinary sense and with respect to a norm in the Sobolev space - which is the condition for the eigenvalues and eigenfunctions of a Hamiltonian to converge to the exact ones. However, as to the speed of convergence the two basis sets (1.1) and (1.2) differ fundamentally. 

Above and throughout this chapter i, for k integer, will stand for the norm in the Sobolev space (space of real functions defined in fl) or H (0) (space of vector or... 

The space of functions // (lZn) is called a Sobolev space. The supindex 1 refers to the fact that the definition of the norm contains only first order derivatives. 

Note that here we only consider functions on the usual three-dimensional coordinate space TZ . The letter L refers to Lebesque integration, a feature that assures that the function spaces are complete (complete normed spaces are also called Banach spaces). We will, however, not go into the detailed mathematics and refer the interested reader to the literature [4]. We just note that for continuous functions the integral is equivalent to the usual (Riemann) integral. Equation (16) defines a norm on the space If and we see from equation (10) that the density belongs to L1. From the condition of finite kinetic energy and the use of a Sobolev inequality one can show that [1]... 

The variational formulation of the stochastic boundary value problem necessitates the introduction of the Sobolev space 77o(72) of functions having generalized derivatives in L D) and vanishing on the boundary dD with norm. 1/2... 

J. Duchon (1977) Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Constructive Theory of Functions of Several Variables, W. Schempp and K. ZeUer (eds.). Springer, Berlin, 85-100. 

Let us consider a semi-Hilbert space valued functions defined on a bounded domain f C with semi-norm s and associated nullspace V. The most well-known example of this setting is the homogeneous Sobolev space Bm with semi-norm - m and nullspace as introduced above. Let V be a finite-dimensional subspace of S with P c V. 

In practice, a semi-norm has to be specified. Instead of the minimum curvature principle used by Briggs, we consider the second order homogeneous Sobolev semi-norm... 

Consider the bivariate homogeneous Sobolev semi-norm (d g x,yy ... 

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