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Register a new userConstrained Spline Smoothing
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Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf, $qgeq 3$, with one additional degree of smoothness) to be of minimal defect while keeping it close to the original function in the ${mathbb L}_p$-(quasi)norm. It is well known that approximating a function by ppfs of minimal defect (splines) avoids introduction of artifacts which may be unrelated to the original function, thus it is always preferable. On the other hand, it is usually easier to construct constrained ppfs with as little requirements on smoothness as possible. Our results allow to obtain shape-preserving splines of minimal defect with equidistant or Chebyshev knots. The validity of the corresponding Jackson-type estimates for shape-preserving spline approximation is summarized, in particular we show, that the ${mathbb L}_p$-estimates, $pge1$, can be immediately derived from the ${mathbb L}_infty$-estimates.
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Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $mathbb{R}^n$ with $k < n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $mathbb{R}^n$, or a solution operator for the transport equation in $mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.
The marriage of recurrent neural networks and neural ordinary differential networks (ODE-RNN) is effective in modeling irregularly-observed sequences. While ODE produces the smooth hidden states between observation intervals, the RNN will trigger a hidden state jump when a new observation arrives, thus cause the interpolation discontinuity problem. To address this issue, we propose the cubic spline smoothing compensation, which is a stand-alone module upon either the output or the hidden state of ODE-RNN and can be trained end-to-end. We derive its analytical solution and provide its theoretical interpolation error bound. Extensive experiments indicate its merits over both ODE-RNN and cubic spline interpolation.
A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh D in R^k, we study the subdivision D obtained by subdividing a maximal cell of D. We give sufficient conditions for the module of splines on D to split as the direct sum of splines on D and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate generalizations.
We introduce the new class of planar Pythagorean-Hodograph (PH) B-Spline curves. They can be seen as a generalization of the well-known class of planar Pythagorean-Hodograph (PH) Bezier curves, presented by R. Farouki and T. Sakkalis in 1990, including the latter ones as special cases. Pythagorean-Hodograph B-Spline curves are non-uniform parametric B-Spline curves whose arc-length is a B-Spline function as well. An important consequence of this special property is that the offsets of Pythagorean-Hodograph B-Spline curves are non-uniform rational B-Spline (NURBS) curves. Thus, although Pythagorean-Hodograph B-Spline curves have fewer degrees of freedom than general B-Spline curves of the same degree, they offer unique advantages for computer-aided design and manufacturing, robotics, motion control, path planning, computer graphics, animation, and related fields. After providing a general definition for this new class of planar parametric curves, we present useful formulae for their construction, discuss their remarkable attractive properties and give some examples of their practical use.
We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $|psi(| abla|) exp(itphi(| abla|)f |_{L^2(w)} leq C|f|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,psi,phi)$. Furthermore, when $w$ is homogeneous, and for certain $(psi,phi)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.
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