Functions In Lp Research Articles

On the uniform convergence of Fourier series to (ψ, β)-derivatives

In terms of the best approximations of a function in the space Lp, the conditions of existence of its (ψ, β)-derivatives and the uniform convergence of Fourier series to them are determined.

Algorithmic Randomness and Fourier Analysis

Suppose 1 < p < ∞. Carleson’s Theorem states that the Fourier series of any function in Lp[−π, π] converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every f ∈ Lp[−π, π] given natural computability conditions on f and p.

Wavelets and Bidemocratic Pairs in Weighted Norm Spaces

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted Lp spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces Lp, p≠2.

On the boundedness of square function generated by the Bessel differential operator in weighted LebesqueLp,αspaces

AbstractIn this paper, we consider the square function(Sf)(x)=(∫0∞|(f⊗Φt)(x)|2dtt)1/2$$\begin{array}{} \displaystyle (\mathcal{S}f)(x)=\left( \int\limits_{0}^{\infty }|(f\otimes {\it\Phi}_{t})\left( x\right) |^{2}\frac{dt}{t}\right) ^{1/2} \end{array} $$associated with the Bessel differential operatorBt=d2dt2+(2α+1)tddt,$\begin{array}{} B_{t}=\frac{d^{2}}{dt^{2}}+\frac{(2\alpha+1)}{t}\frac{d}{dt}, \end{array} $α&gt; −1/2,t&gt; 0 on the half-line ℝ+= [0, ∞). The aim of this paper is to obtain the boundedness of this function inLp,α,p&gt; 1. Firstly, we provedL2,α-boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) andLp,α,p&gt; 1 boundedness are proved by taking into account vector-valued functions.

Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp

We give new formulae for analytic balayage of measures supported on subsets of Carathéodory compact sets in the complex plane, and consider a related problem of description of badly approximable functions in Lp(T).

The Lp,q-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces Lp,q(ℝd+1)

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, $$\sum\nolimits_{k \in \mathbb{Z}^{d + 1} } {\left| {\hat \varphi (\xi + 2\pi k)} \right|^2 > 0}$$ . Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results.

Approximation by Θ-Means of Walsh–Fourier Series

We discuss the behaviour of the Θ-means of Walsh series of a function in L p (1 ≤ p ≤ ∞). We investigate the rate of the approximation by this means, in particular, in Lip(α, p), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞ by L p we mean C, the class of the continuous functions. Our main theorems give a common generalization of two results of Moricz and Siddiqi on Norlund means [11] and Moricz and Rhoades on weighted means [9].

Modular inequalities for the maximal operator in variable Lebesgue spaces

A now classical result in the theory of variable Lebesgue spaces due to Lerner (2005) is that a modular inequality for the Hardy–Littlewood maximal function in Lp(⋅)(Rn) holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality ∫ΩMf(x)p(x)dx⩽c1∫Ω|f(x)|q(x)dx+c2,where c1,c2 are non-negative constants and Ω is any subset of Rn. As a corollary we get sufficient conditions for the modular inequality ∫Ω|Tf(x)|p(x)dx⩽c1∫Ω|f(x)|q(x)dx+c2,where T is any operator that is bounded on Lp(Ω), 1<p<∞.

Про швидкiсть збiжностi для оцiнки iмпульсної перехiдної функцiї у просторi Lp(T)

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space Lp(T).

Rate of approximation by $q$ -Durrmeyer operators in $L_{p}([0,1])$ , $1\leq p\leq\infty$

We obtain global rates of approximation by q-Durrmeyer operators Dn,q(f;x) for the functions in the class Lp([0,1]),1≤p≤∞. First, rates of approximation in terms of the norms of f and f' and in terms of the ordinary modulus of smoothness are obtained. Subsequently, we obtain rates of approximation in terms of the generalized modulus of smoothness ωφ(f,δ).

The geometry of two-valued subsets of Lp -spaces

Abstract Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q &gt; p. Such sets preclude the existence of certain types of isometry into L p -spaces.

Best Simultaneous Local Approximation in the Lp Norms

ABSTRACTWe study the behavior of the best simultaneous approximation to two functions from a convex set in Lp spaces, 2<p<∞, on a finite union of intervals when its measure tends to zero. In particular, we give sufficient conditions over the differentiability of two functions to assure existence of the best simultaneous local approximation from the class of algebraic polynomials of a fixed degree. These conditions are weaker than the ordinary differentiability given in previous works. More precisely, we consider differentiable functions in the sense Lp.

Lipoprotein(a) and its role in inflammation, atherosclerosis and malignancies

Lipoprotein (a) (Lp(a)) is a modified low-density lipoprotein (LDL) particle with an additional specific apolipoprotein (a), covalently attached to apolipoprotein B‑100 of LDL by a single thioester bond. Increased plasma Lp(a) level is a genetically determined, independent, causal risk factor for cardiovascular disease.The precise quantification of Lp(a) in plasma is still hampered by mass-sensitive assays, large particle variation, poor standardization and lack of assay comparability.The physiological functions of Lp(a) include wound healing, promoting tissue repair and vascular remodeling. Similarly to other lipoproteins, Lp(a) is also susceptible for oxidative modifications, leading to extensive formation of pro-inflammatory and pro-atherogenic oxidized phospholipids, oxysterols, oxidized lipid-protein adducts in Lp(a) particles, that perpetuate atherosclerotic lesion progression and intima-media thickening through induction of M1-macrophages, inflammation, autoimmunity and apoptosis. The oxidation-specific epitopes of modified lipoproteins are major targets of pre-immune, natural IgM antibodies, that may attenuate the pro-inflammatory and pro-atherogenic effects of Lp(a).Although the data are still insufficient, recent studies suggest a potential anti-neoplastic role of Lp(a).

Analytic representations of sequences in Lp spaces, 1 ≤ p &lt; ∞

In this paper we consider a sequence of functions in Lp(R), 1 ? p &lt; ? and, in the second part, we include a sequence of real analytic functions without real roots. We obtain several results regarding their convergence or the convergence of the sequence of their analytic representations. We, also, give results about the analytic representation of the product of the boundary functions and other additional results.

Paley–Wiener theorem for functions in Lp(ℝn)

ABSTRACTIn this paper we examine necessary and sufficient conditions on the sequences of norm of the derivatives of functions in such that their spectrum (the support of their Fourier transform) is contained in a given compact set in .

Approximation of the Quadratic Partial Sums of Double Walsh–Kaczmarz–Fourier Series by Nörlund Means

We discuss the Norlund means of quadratic partial sums for theWalsh–Kaczmarz–Fourier series of a function in L p . The rate of approximation by these means is investigated, in particular, in Lip( $$ \alpha $$ , p), where $$ \alpha $$ > 0 and 1 ≤ p ≤ ∞. For p = ∞, the set L p turns into the collection of continuous functions C. Our main theorem states that the approximation behavior of these two-dimensional Walsh–Kaczmarz–Norlund means is as good as the approximation behavior of the one-dimensional Walsh and Walsh–Kaczmarz–Norlund means. Earlier, the results for one-dimensional Norlund means of the Walsh–Fourier series were obtained by M´oricz and Siddiqi [J. Approxim. Theory, 70, No. 3, 375–389 (1992)] and Fridli, Manchanda, and Siddiqi [Acta Sci. Math. (Szeged), 74, 593–608 (2008)]. For one-dimensionalWalsh–Kaczmarz–Norlund means, the corresponding results were obtained by the author [Georg. Math. J., 18, 147–162 (2011)]. The case of two-dimensional trigonometric systems was studied by Moricz and Rhoades [J. Approxim. Theory, 50, 341–358 (1987)].

Lipoprotein (a): truly a direct prothrombotic factor in cardiovascular disease?

Elevated plasma concentrations of lipoprotein (a) [Lp(a)] have been determined to be a causal risk factor for coronary heart disease, and may similarly play a role in other atherothrombotic disorders. Lp(a) consists of a lipoprotein moiety indistinguishable from LDL, as well as the plasminogen-related glycoprotein, apo(a). Therefore, the pathogenic role for Lp(a) has traditionally been considered to reflect a dual function of its similarity to LDL, causing atherosclerosis, and its similarity to plasminogen, causing thrombosis through inhibition of fibrinolysis. This postulate remains highly speculative, however, because it has been difficult to separate the prothrombotic/antifibrinolytic functions of Lp(a) from its proatherosclerotic functions. This review surveys the current landscape surrounding these issues: the biochemical basis for procoagulant and antifibrinolytic effects of Lp(a) is summarized and the evidence addressing the role of Lp(a) in both arterial and venous thrombosis is discussed. While elevated Lp(a) appears to be primarily predisposing to thrombotic events in the arterial tree, the fact that most of these are precipitated by underlying atherosclerosis continues to confound our understanding of the true pathogenic roles of Lp(a) and, therefore, the most appropriate therapeutic target through which to mitigate the harmful effects of this lipoprotein.

Density of smooth functions in variable exponent Sobolev spaces

We show that if p−≥2, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space W1,p(⋅)(Rn), is that the Riesz potentials of compactly supported functions of Lp(⋅)(Rn), are also elements of Lp(⋅)(Rn). Using this result we then prove that the above density holds if (i) p−≥n or if (ii) 2≤p−<n and p+<np−n−p−. Moreover our result allows us to give an alternative proof, for the case p−≥2, that the local boundedness of the maximal operator and hence local log-Hölder continuity imply the density of smooth functions with compact support, in the variable exponent Sobolev space W1,p(⋅)(Rn).

Boundedness of integral operators on decreasing functions

We continue the study of the boundedness of the operatoron the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p &gt; 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

A semi-group formula for the Riesz potentials

The purpose of this article is to establish a semi-group formula for the Riesz potentials of Lp-functions. As preparations, we study the Lizorkin space Φ(Rn) and investigate integral estimates of the Riesz potentials of functions in the spaces Lp:r,s(Rn).