The $C^m$ Norm of a Function with Prescribed Jets I

Abstract

We prove a variant of the classical Whitney extension theorem, in which the C m -norm of the extending function is controlled up to a given, small percentage error. IntroductionHere and in [16], we compute the least possible (infimum) C m norm of a function F having prescribed Taylor polynomials at N given points of R n . Moreover, given > 0, we exhibit such an F, whose C m norm is within of the least possible. Our computation consists of an algorithm, to be implemented on an (idealized) digital computer. The algorithm works, thanks to a variant of the classical Whitney extension theorem, in which we control the C m norm of the extending function up to an percentage error. This paper gives the variant of Whitney's theorem, while [16] presents the algorithm and the rest of the mathematics behind it. The number of operations used by our algorithm is C( )N log N, where N is the number of given points, and C( ) grows rapidly as tends to zero.To state our results precisely, we set up notation. Fix m, n ≥ 1. We pick a norm on C m (R n ), subject to restrictions to be spelled out in the next section. We write F C m (R n ) to denote the norm of F. Given x ∈ R n and F ∈ C m (R n ), we write J x (F) to denote the m th order Taylor polynomial of F at x. Thus, J x (F) belongs to P, the vector space of all (real) m th degree polynomials on R n . Let E ⊂ R n . We write #(E) for the number of points in E. (If E is infinite, then #(E) = +∞.) A Whitney field on E is a family P = (P x ) x∈E of polynomials P x ∈ P, indexed by x ∈ E. If P = (P x ) x∈E is a Whitney field and S ⊂ E is a subset, then in an obvious way we can define the restriction P| S of P to S.

An example related to Whitney extension with almost minimal $C^m$ norm

“…m norm 425 Conjecture 1.2 admits a positive answer in the case m = 1, for a large variety of C 1 norms, as is explained in [18]. However, in this note we demonstrate that Conjecture 1.2 is false already in the first non-trivial case m = n = 2, for most reasonable choices of a C 2 norm.…”

“…Motivated by the practical problem of multi-variate interpolation, an attempt to study the C m norm on a more accurate level was initiated by the first named author in [18,19] 1 . Among the positive results obtained in these works, is an analog of the classical Whitney theorem for jets (see [27] or [26, Section VI]), with an accurate control of the C m norm.…”

“…Among the positive results obtained in these works, is an analog of the classical Whitney theorem for jets (see [27] or [26, Section VI]), with an accurate control of the C m norm. The investigations in [18,19] take into account various possible definitions of C m norms, in addition to the definition (1.1). For instance, the results in [18,19] The latter definition of a C m norm is exactly rotationally-invariant, and hence, perhaps, is slightly more natural than the definition (1.1).…”

“…The investigations in [18,19] take into account various possible definitions of C m norms, in addition to the definition (1.1). For instance, the results in [18,19] The latter definition of a C m norm is exactly rotationally-invariant, and hence, perhaps, is slightly more natural than the definition (1.1). Of course, the definitions (1.1) and (1.2) are equivalent up to a constant depending only on m and n. The finer analysis in [18,19] has led to the formulation of the following optimistic conjecture.…”

Designing multidirectional energy splitters and topological valley supernetworks

“…• Chirality of valley states and phase matching.-Unique to topological valley modes is presence of a favoured chirality for edge modes [10,47,49]. The lack of coupling of modes with opposite chirality has been shown in [32].…”