Finite Energy Solutions of Maxwell''s Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds

We consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of L^p forms in Lipschitz domains, we show that both are well posed provided that 2−<p<2+, for some >0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

The n-width of the unit ball of H

Let E be a compact subset of the open unit disc Δ and let H^q be the Hardy space of analytic functions f on Δ for which stf¦^q has a harmonic majorant. We determine the value of the Kolmogorov, Gel'fand, and linear n-widths in L^p(E, μ) of the restriction to E of the unit ball of H^q when p q or when 1 q < p < ∞ and E is “small”.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

Product properties of Hilbert transforms

Let fεL^p(R), gεL^q(R) with 1<p<∞, 1<q<∞ and let Hf, Hg be their respective Hilbert transforms. We give a simple proof of the identity Hf · Hg − f · G = H(f · Hg + g · Hf) a.e. and of its inverse in the case (1/p) + (1/q) 1 which includes the cases already considered by Cossar and Tricomi. We next derive applications, especially to boundary values of analytic functions.     

Étale slices for representation varieties in characteristic p

Let R(F, G) be the variety of representations of a finitely generated group F into a connected reductive algebraic group G, and let C(F, G) be the variety of closed conjugacy classes of representations. We examine the question of whether an étale slice for the conjugation action of G exists through a representation ρR(F, G) when the ground field k has characteristic p > 0. We show that an étale slice through ρ may exist for the action of an enlarged group Ĝ, even when there is no étale slice for the G-action. As an application, we generalise a result known to hold in characteristic zero, which expresses the tangent space to C(F, G) at the conjugacy class of a suitable representation ρ as a subspace of the 1-cohomology H¹ (F, %plane1D;524;(ρ)) of an F-module %plane1D;524;(ρ). A similar result holds in characteristic p, but with H¹ (F%plane1D;524;(ρ)) replaced by a quotient of H¹ (F%plane1D;524;(ρ)).     

A general framework of compactly supported splines and wavelets

Let {V_k} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L²( ). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ε Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W_k of V_k, . In particular, the minimally supported ψ ε Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L² functions from the “spline” spaces V_k and “wavelet” spaces W_k, k . A very general duality principle, which yields the dual bases of both {(·−j):j and {η(·−j):j } for any η ε Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both and ψ have linear phases. Hence, “non-symmetric” and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ_m and ψ_m have linear phases, but the odd-order B-wavelet only has generalized linear phases.     

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets provide an orthonormal basis for L²(R²) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x₁ cos θ+x₂ sin θ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B^1/p_p, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions.     

Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2).     

Weighted Hardy and Opial-type inequalities

An integral condition on weights u and v is given which is equivalent to the boundedness of the Hardy operator between the weighted Lebesgue spaces L_u^p and L_v^q with 0 < q < 1 < p < ∞. The Hardy inequalities are applied to give easily verified weight conditions which imply inequalities of Opial type.     

Analytic approximation of matrix functions in

We consider the problem of approximation of matrix functions of class L^p on the unit circle by matrix functions analytic in the unit disk in the norm of L^p, 2≤p<∞. For an m×n matrix function Φ in L^p, we consider the Hankel operator , 1/p+1/q=1/2. It turns out that the space of m×n matrix functions in L^p splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of H_Φ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of L^p. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.     

Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk

Let f be an integrable function on the unit disk. The Hankel operator H_f is densely defined on the Bergman space A^p by H_fg = fg − P(fg), where g is a bounded analytic function and P is the Bergman projection (orthogonal projection from L² to A²) extended to L¹ via its integral formula. In this paper, the functions f for which H_f extends to a bounded operator from A^p to L^p are characterized, 1 < p < ∞. Also characterized are the functions f for which H_f extends to a compact or Schatten class operator on A². The proofs can be extended to handle any smoothly bounded domain in C in place of the unit disk.     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

Remarks Concerning Completeness of Translates in Function Spaces

This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k ₊ are complete in the spaces C^l₀( ), L^p( ), W^l_p( ), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

A Turán type inequality for rational functions with prescribed poles

Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include L^p spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.     

Heat Diffusion on Homogeneous Trees

Let X be a homogeneous tree. We study the heat diffusion process associated with the nearest neighbour isotropic Markov operator on X. In particular it is shown that the heat maximal operator is weak type (1, 1) and strong type (p, p), for every 1 < p < ∞. We estimate the asymptotic behaviour of the heat maximal function. Moreover, we introduce a family of H^p spaces on X. It is proved that H^p=l^p(X) for 1 < p < ∞ and is conjectured that H^p for p less than 1, is trivial.     

A generalized mean value inequality for subharmonic functions

If u ≥ 0 is subharmonic on a domain Ω in ⁿ and 0 < p < 1, then it is well-known that there is a constant C(n,p) ≥ 1 such u(x)^p ≤ C)n,p) MV )u^p,B(x,r)) for each ball B(x,r)) Ω. We show more generally that a similar result holds for functions ψ : ₊ → ₊ may be any surjective, concave function whose inverse ψ⁻¹ satisfies the Δ₂-condition.