Lebesgue Differentiation Theorem Research Articles

High-order adaptive multiresolution wavelet upwind schemes for hyperbolic conservation laws

We present a novel system of high-order adaptive multiresolution wavelet collocation upwind schemes, implemented in a meshfree framework, for solving hyperbolic conservation laws. Our approach constructs asymmetrical wavelet bases with interpolation properties to achieve the upwind property and address nonlinearities in hyperbolic problems. An adaptive algorithm based on multiresolution analysis in wavelet theory is designed to capture moving shock waves and identify new localized steep regions. To suppress the Gibbs phenomenon, we propose an integration average reconstruction method based on the Lebesgue differentiation theorem. These numerical techniques enable the wavelet collocation upwind scheme to provide a general framework for devising satisfactory adaptive wavelet upwind methods with high-order accuracy. We perform several benchmark tests for 1D hyperbolic problems to verify the accuracy and efficiency of the proposed wavelet schemes. Notably, our wavelet collocation upwind schemes successfully solve the two interacting blast waves problem when discretizing the spatial derivatives in the physical space directly. This is in contrast to classical high-order schemes that usually involve transforming variables back and forth between physical and characteristic spaces with many local projections. Our results indicate the robustness, stability, and efficiency of our proposed schemes for strong shock wave interaction problems.

The metric-valued Lebesgue differentiation theorem in measure spaces and its applications

We prove a version of the Lebesgue differentiation theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon–Nikodým property.

Density topologies for strictly positive Borel measures

The aim of this paper is to describe density topologies generated by Borel complete regular measures. For any such measure μ a generalization of the classical Lebesgue differentiation theorem is proved here. Using it, we define a topology Tμ which is some kind of abstract density topology and, simultaneously, a natural generalization of the classical density topology Td.We focus on separation axioms of topologies Tμ. We show that considered topologies are regular and not normal. In some cases we prove that Tμ gives a Tychonoff space. We also consider homeomorphisms between topologies Tμ. It is shown that for any atomless measure ν the topology Tν is homeomorphic to Td. However, there are measures μ generating topologies quite different than Td, for example separable and not connected. Some cases deliver us examples of spaces which are not Lindelöf but they are separable.

Generalized Differentiability of Continuous Functions

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

A median approach to differentiation bases

We study a version of the Lebesgue differentiation theorem in which the integral averages are replaced with medians over Busemann–Feller differentiation bases. Our main result gives several characterizations for the differentiation property in terms of the corresponding median maximal function. As an application, we study pointwise behaviour in Besov and Triebel–Lizorkin spaces, where functions are not necessarily locally integrable. Most of our results apply also for functions defined on metric measure spaces.

The Lebesgue differentiation theorem revisited

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on Rn in terms of approximate continuity associated to an expansive linear map.

Norms supporting the Lebesgue differentiation theorem

A version of the Lebesgue differentiation theorem is offered, where the [Formula: see text] norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

Time-Inconsistent Stochastic LinearrQuadratic Control: Characterization and Uniqueness of Equilibrium

In this paper, we continue our study on a general time-inconsistent stochastic linear-quadratic (LQ) control problem originally formulated in Hu, Jin and Zhou (2012). We derive a necessary and sufficient condition for equilibrium controls via a flow of forward-backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in Hu, Jin and Zhou (2012) is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.

Time-Inconsistent Stochastic Linear-Quadratic Control: Characterization and Uniqueness of Equilibrium

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.

A Short Note on Weak Estimation of Sharp Function

In paper [1] Ahmad et al. investigated the use of sharp function, known from functional analysis, in image processing. The sharp function gives a measure of variations of a function and can be used as an edge detector [2]. We extend the classical notion of sharp function to prove the classical Lebesgue differentiation theorem and Marcinkiewicz theorem for the sublinear operator T(x;y).

Schnorr randomness and the Lebesgue differentiation theorem

Schnorr randomness and the Lebesgue differentiation theorem

Generalised Gagliardo–Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for $${1 \leq q < p < \infty}$$ and $${s \geq 0}$$ with s > n(1/2 − 1/p), if $${f \in L^{q,\infty} (\mathbb{R}^n) \cap \dot{H}^s (\mathbb{R}^n)}$$ , then $${f \in L^p(\mathbb{R}^n)}$$ and there exists a constant c p,q,s such that $$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$ where 1/p = θ/q + (1−θ)(1/2−s/n). In particular, in $${\mathbb{R}^2}$$ we obtain the generalised Ladyzhenskaya inequality $${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$$ .We also show that for s = n/2 and q > 1 the norm in $${\| f \|_{\dot{H}^{n/2}}}$$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon–Zygmund decompositions.

Integration in quasi-Banach spaces and the fundamental theorem of calculus

We make a general approach to integrability and its interplay with differentiability in quasi-Banach spaces. This endeavor demands studying first the defects of Bochner and Riemann integration in the setting of p-Banach spaces when p<1. The conclusion will be that the local convexity is a necessary (and sufficient) condition of the space for the integral operator to work in the expected way. On the positive side, we obtain a criterion for Riemann integrability of quasi-Banach valued maps based on an approximation method by polynomial functions. Finally, with an eye to finding a class of functions whose integral interacts well with differentiation, we give sufficient conditions that guarantee the fulfillment of the fundamental theorem of calculus, and prove the Lebesgue differentiation theorem for the integral in the sense of Vogt.

Computability, Complexity and Randomness

This special issue of Theory or Computing Systems consists of papers associated with the 6th Annual CCR conference held in beautiful Cape Town, 31st January– February 4th 2011. The conference series is devoted to issues around algorithmic information theory, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. In 2011, the conference was co-located with the 8th Annual Computability and Analysis conference. This co-location reflects the increasing interactions between computable analysis and algorithmic randomness through effective analysis of almost everywhere behavior in analysis such as the Lebesgue Differentiation Theorem and things like Brownian motion. Both conferences were admirably overseen by Vasco Brattka and his program committees. The last 10–15 years has seen huge progress in the areas represented by CCR with two long monographs in the area, recognition at the International Congress of Mathematicians, and a thoroughly thriving research community. Whilst the CCR conference uses a model which does not have an actual proceedings at the meeting, the papers in this special issue reflect the meeting’s business. These papers reflect the high caliber of both the researchers and the intellectual merit of the investigations. The work spans issues from applications in biology, computer science to fundamental issues about Kolmogorov complexity. The papers all went through the full Theory of Computing Systems journal reviewing process and several of the initial submissions were rejected. Enjoy.

The best constant approximant operators in Lorentz spaces [formula omitted] and their applications

In the present article we extend the best constant approximant operator from Lorentz spaces Γ p , w to Γ p − 1 , w for any 1 < p < ∞ and w ≥ 0 a locally integrable weight function, and from Γ 1 , w to the space of all measurable functions L 0 . Then we establish several properties of the extended best constant approximant operator and finally, we prove a generalized version of the Lebesgue Differentiation Theorem in L 0 .

A computational aspect of the Lebesgue differentiation theorem

Given an L1 -computable function, f , we identify a canonical represen- tative of the equivalence class of f , where f and g are equivalent if and only if R jf gj is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-L¨ random points in Euclidean space. 2000 Mathematics Subject Classification 03D80 (primary); 26A24 (secondary)

Differentiability properties of Orlicz-Sobolev functions

In this paper we are concerned with the pointwise behaviour of functions in certain classes of weakly differentiable functions. The ancestor of all modern results dealing with pointwise properties of nonsmooth functions is certainly the Lebesgue differentiation theorem, which asserts that if ft is an open subset of R ~, n > l , and u is a locally integrable function in f~, then lim~_~0+ 3CB,.(x) u(y) dy exists and is finite for a.e. xEf~, and

Vitali Coverings and Lebesgue's Differentiation Theorem

The standard techniques used to prove the Lebesgue differentiation theorem (that monotonic functions are a.e.~differentiable) are presented in an unusual way that reveals more about their nature and allows greater generality.

The Lebesgue Differentiation Theorem via the Rising Sun Lemma

A complete version of Lebesgue's differentiation theorem, including the image of the exceptional set, is proved in an elementary way.

A Note on the Lebesgue Differentiation Theorem in Spaces of Homogeneous Type

We prove that the Lebesgue differentiation theorem holds in the general setting of spaces of homogeneous type if the balls are subspaces of homogeneous type.