Functions In Lp Research Articles

Constrained Lp-Approximation by Generalized n-Convex Functions Induced by ECT-Systems

The problem of finding a best Lp-approximation (1 ≤ p < ∞) to a function in Lp from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L1 are investigated. Some special results for generalized monotone and convex cases are obtained.

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1&lt;P&lt;+∞

Let Γ be a regular curve and Lp(Γ),1<p<+∞, be the class of all complex-valued functions f defined on Γ which are such that |f|p is integrable in sense of Lebesgue. In this work, we define the kth p-Faber polynomial Fk.p(z), the kth p-Faber principle part ≈Fk.p(1/z) for Γ, and defined the nth p-Faber-Laurent rational function Rn,p(f, z) and p-generalized modulus of continuity Ωp of a function f of Lp(Γ). We investigate some properties of Fk.p(z) and ≈Fk.p(1/z). And then we prove a direct theorem characterizing the degree of approximation with respect to Ωp in the mean of functions of Lp(Γ) by the rational functions Rn.p(.,z).

Approximation by weighted means of Walsh‐Fourier series

We study the rate of approximation to functions in Lp and, in particular, in Lip(α, p) by weighted means of their Walsh‐Fourier series, where α &gt; 0 and 1 ≤ p ≤ ∞. For the case p = ∞, Lp is interpreted to be CW, the collection of uniformly W continuous functions over the unit interval [0, 1). We also note that the weighted mean kernel is quasi‐positive, under fairly general conditions.

Dual Cones, Constrained n-Convex Lp-Approximation, and Perfect Splines

A generating basis and the dual cone of n-convex functions satisfying certain constraints are derived. As applications, the existence and characterization of a best Lp-approximation (1 ≤ p < ∞) from such subcones to a function in Lp are established. The relationship between a best L1-approximation and perfect splines is developed under certain conditions.

The Toeplitz theorem and its applications to approximation theory and linear PDEs

We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection Φ \Phi of compactly supported functions in L p ( R s ) ( 1 ⩽ p ⩽ ∞ ) {L_p}({\mathbb {R}^s})\quad (1 \leqslant p \leqslant \infty ) , we consider the shift-invariant space S S generated by Φ \Phi and the family ( S h : h &gt; 0 ) ({S^h}:h &gt; 0) , where S h {S^h} is the h h -dilate of S S . We prove that ( S h : h &gt; 0 ) ({S^h}:h &gt; 0) provides L p {L_p} -approximation order r r only if S S contains all the polynomials of total degree less than r r . In particular, in the case where Φ \Phi consists of a single function φ \varphi with its moment ∫ φ ≠ 0 \int {\varphi \ne 0} , we characterize the approximation order of ( S h : h &gt; 0 ) ({S^h}:h &gt; 0) by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.

Inverse theorem for best polynomial approximation in 𝐿_{𝑝},0&lt;𝑝&lt;1

A direct theorem for best polynomial approximation of a function in L p [ − 1 , 1 ] , 0 &gt; p &gt; 1 {L_p}[ - 1,1],\;0 &gt; p &gt; 1 , has recently been established. Here we present a matching inverse theorem. In particular, we obtain as a corollary the equivalence for 0 &gt; α &gt; k 0 &gt; \alpha &gt; k between E n ( f ) p = O ( n − α ) {E_n}{(f)_p} = O({n^{ - \alpha }}) and ω φ k ( f , t ) p = O ( t α ) \omega _\varphi ^k{(f,t)_p} = O({t^\alpha }) . The present result complements the known direct and inverse theorem for best polynomial approximation in L p [ − 1 , 1 ] , 1 ⩽ p ⩽ ∞ {L_p}[ - 1,1],\;1 \leqslant p \leqslant \infty . Analogous results for approximating periodic functions by trigonometric polynomials in L p [ − π , π ] , 0 &gt; p ⩽ ∞ {L_p}[ - \pi ,\pi ],0 &gt; p \leqslant \infty , are known.

Pointwise convergence of wavelet expansions

In this note we announce that under general hypotheses, wavelet-type expansions (of functions in L p {L^p} , 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , in one or more dimensions) converge pointwise almost everywhere, and identify the Lebesgue set of a function as a set of full measure on which they converge. It is shown that unlike the Fourier summation kernel, wavelet summation kernels P j {P_j} are bounded by radial decreasing L 1 {L^1} convolution kernels. As a corollary it follows that best L 2 {L^2} spline approximations on uniform meshes converge pointwise almost everywhere. Moreover, summation of wavelet expansions is partially insensitive to order of summation. We also give necessary and sufficient conditions for given rates of convergence of wavelet expansions in the sup norm. Such expansions have order of convergence s if and only if the basic wavelet ψ \psi is in the homogeneous Sobolev space H h − s − d / 2 H_h^{ - s - d/2} . We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order δ≥0 for suitable test functions on ℝN are defined via the Fourier transform by\((S_R^\delta f)(\xi ) = (1 - |\xi |^2 /R^2 )^\delta + \hat f(\xi )\). We show that the means of the critical index\(\delta = \frac{N}{p} - \frac{{N + 1}}{2},1< p< \frac{{2N}}{{N + 1}}\), do not mapLp,∞(ℝN) intoLp,∞(ℝN), but they map radial functions ofLp,∞(ℝN) intoLp,∞(ℝN). Moreover, iff is radial and in theLp,∞(ℝN) closure of test functions,SRδf(x) converges, asR→+∞, tof(x) in norm and for almost everyx in ℝN. We also observe that the means of the function|x|−N/p, which belongs toLp,∞(ℝN) but not to the closure of test functions, converge for nox.

A structural assessment of the apo[a] protein of human lipoprotein[a

Apolipoprotein[a], the highly glycosylated, hydrophilic apoprotein of lipoprotein[a] (Lp[a]), is generally considered to be a multimeric homologue of plasminogen, and to exhibit atherogenic/thrombogenic properties. The cDNA-inferred amino acid sequence of apo[a] indicates that apo[a], like plasminogen and some zymogens, is composed of a kringle domain and a serine protease domain. To gain insight into possible positive functions of Lp[a], we have examined the apo[a] primary structure by comparing its sequence with those of other proteins involved in coagulation and fibrinolysis, and its secondary structure by using a combination of structure prediction algorithms. The kringle domain encompasses 11 distinct types of repeating units, 9 of which contain 114 residues. These units, called kringles, are similar but not identical to each other or to PGK4. Each apo[a] kringle type was compared with kringles which have been shown to bind lysine and fibrin, and with bovine prothrombin kringle 1. Apo[a] kringles are linked by serine/threonine- and proline-rich stretches similar to regions in immunoglobulins, adhesion molecules, glycoprotein Ib-alpha subunit, and kininogen. In comparing the protease domains of apo[a] and plasmin, apo[a] contains a region between positions 4470 and 4492 where 8 substitutions, 9 deletions, and 1 insertion are apparent. Our analysis suggests that apo[a] kringle-type 10 has a high probability of binding to lysine in the same way as PGK4. In the only human apo[a] polymorph sequenced to date, position 4308 is occupied by serine, whereas the homologous position in plasmin is occupied by arginine and is an important site for proteolytic cleavage and activation. An alternative site for the proteolytic activation of human apo[a] is proposed.

Weighted Generalized Hardy Inequalities for Nonincreasing Functions

AbstractThe nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

Criterion for holomorphic continuability of the functions of class LP defined on a portion of the Shilov boundary of a circular strongly starlike domain

Criterion for holomorphic continuability of the functions of class LP defined on a portion of the Shilov boundary of a circular strongly starlike domain

Adapted sets of measures and invariant functionals on 𝐿^{𝑝}(𝐺)

Let G G be a locally compact group. If G G is compact, let L 0 p ( G ) L_0^p(G) denote the functions in L p ( G ) {L^p}(G) having zero Haar integral. Let M 1 ( G ) {M^1}(G) denote the probability measures on G G and let P 1 ( G ) = M 1 ( G ) ∩ L 1 ( G ) {\mathcal {P}^1}(G) = {M^1}(G) \cap {L^1}(G) . If S ⊆ M 1 ( G ) S \subseteq {M^1}(G) , let Δ ( L p ( G ) , S ) \Delta ({L^p}(G),S) denote the subspace of L p ( G ) {L^p}(G) generated by functions of the form f − μ ∗ f f - \mu \ast f , f ∈ L p ( G ) f \in {L^p}(G) , μ ∈ S \mu \in S . If G G is compact, Δ ( L p ( G ) , S ) ⊆ L 0 p ( G ) \Delta ({L^p}(G),S) \subseteq L_0^p(G) . When G G is compact, conditions are given on S S which ensure that for some finite subset F F of S S , Δ ( L p ( G ) , F ) = L 0 p ( G ) \Delta ({L^p}(G),F) = L_0^p(G) for all 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty . The finite subset F F will then have the property that every F F -invariant linear functional on L p ( G ) {L^p}(G) is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , conditions are given upon G G , and upon subsets S S of M 1 ( G ) {M^1}(G) whose elements satisfy certain growth conditions, which ensure that L p ( G ) {L^p}(G) has discontinuous, S S -invariant linear functionals. The results are applied to show that for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty , L p ( R ) {L^p}(\mathbb {R}) has an infinite, independent family of discontinuous translation invariant functionals which are not P 1 ( R ) {\mathcal {P}^1}(\mathbb {R}) -invariant.

Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces

According to the well-known theorem of Fatou, the harmonic extension u to the upper half-space R~_ +x of a function f in LP(Rn), 1 1. In this paper we study the analogous question in the unit ball B of C'. Recall that the correct analogue of Fatou's theorem for B is the theorem of Kor/myi [K] stating that invariant harmonic extensions (Poisson-Szeg6 integrals) of functions in LP(S), 1 < p < oo, have "admissible" limits at almost every point of the unit sphere S. In particular, this is true for holomorphic functions in the Hardy space HP(B). As is well known, Korfinyi's admissible regions allow parabolic tangential approach along the complex-tangential directions. These admissible regions are best possible as far as the order of tangency to the boundary is concerned, even if only bounded holomorphic functions are considered, as shown in [H-S] (however, they are not best possible in a stronger sense [Su 1]). By analogy with the real-variable situation, it seems reasonable to expect Kor/myi's regions not to be optimal for holomorphic functions whose (admissible) boundary values are, in some sense, Bessel-type potentials of LP(S) functions. Thus we consider holomorphic functions in the unit ball which are given by "fractional Cauchy integrals" of LP(S) functions. These are obtained by modifying the exponent in the denominator of the Cauchy kernel of B, a procedure suggested by the

The duals of harmonic Bergman spaces

In this paper we show that for Ω \Omega , a starlike Lipschitz domain, the dual of the space of harmonic functions in L p ( Ω ) {L^p}(\Omega ) need not be the harmonic functions in L q ( Ω ) {L^q}(\Omega ) , where 1 / p + 1 / q = 1 1/p + 1/q = 1 . We show that, as a consequence, the harmonic Bergman projection for Ω \Omega need not extend to a bounded operator on L p ( Ω ) {L^p}(\Omega ) for all 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty . The duality result is a partial answer to a question of Nakai and Sario [9] posed initially in the Proceedings of the London Mathematical Society in 1978. We treat the duality question as a biharmonic problem, and our result follows from the failure of uniqueness for the biharmonic Dirichlet problem in domains with sharp intruding corners.

The Conjugate Function on the Finite Dimensional Torus

AbstractWe consider the group Ta, its group of characters Za, and an arbitrary order P on Za. For x ∊ Za, let sgnpx be 1, - 1 , or 0 according as x € P\{0}, x € (-P)\{0}, or X = 0. For f in Lp(Ta), 1 &lt; p &lt; ∞, it is known that there is a function in Lp(Ta) such that for all X in Za. Summability methods for are also available. In this paper, we obtain summability methods for that apply for in L1(Ta), and we show how various properties of can be derived from our construction.

Boundary Regularity for Harmonic Hardy-Sobolev Spaces

Two scales of harmonic Hardy-Sobolev spaces are introduced and their boundary regularity is studied. Both scales impose conditions on derivatives of harmonic functions in a fixed direction. In one case, they are required to have bounded P-means, while in the other, they are required to have non-tangentialmaximal functions in Lp. The results include embedding in Lipschitz spaces, as well as into spaces of continuous functions and spaces of bounded and vanishing mean oscillation. In particular, real variable versions of the theorem of Privalov on analytic functions with absolutely continuous boundary values are proved.

Band-limited functions: 𝐿^{𝑝}-convergence

We consider the set B p ( Ω ) {B_p}(\Omega ) (functions of L p ( R ) {L^p}({\mathbf {R}}) whose Fourier spectrum lies in [ − Ω , + Ω ] [ - \Omega , + \Omega ] ). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if 4 / 3 &gt; p &gt; 4 4/3 &gt; p &gt; 4 . The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inL p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2.

Approximation of periodic functions in LP by linear positive methods and multiple moduli of continuity

If the operator A is positive, i.e., K(t) _> 0, then it is known that in the case of p = i, estimate (3) is good enough, in the sense that its application leads to the precise value of the deviation on various classes of functions. However, for the values p ~ i, = estimate (3) is already rough, which is related to the strict convexity of the corresponding spaces Lp. Our goal is to obtain a more precise than (3) estimate of approximation by positive operators, which would reduce to the equality for a sufficiently wide class of functions. In doing so, we shall use the following variant of the Riesz-Thorin theorem on complex interpolation of operators in the spaces Lp with mixed norm, contained in [i]. Let LDr (i < p, r < ~) be the space of 2~-periodic in each argument complex functiorls g(x, t) such that

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in Lp (IR+, xα) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, α=0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1<p<∞.