Functions In Lp Research Articles

New methods for localizing discontinuities of a noisy function

For a problem of localizing singularities (discontinuities of the first kind) of a noisy function in Lp (1 ≤ p < ∞), new classes of regularizing methods are constructed. The methods determine the number of discontinuities and approximate their positions. The upper and lower bounds of the localizing singularities and the separability threshold are also obtained. It is proved that the methods are order-optimal by accuracy and separability on some classes of functions with discontinuities.

Interpolating Operators for Multiapproximation

Problem statement: There are no simple definitions of operators for be st multiapproximation and best one sided multiapproximation which work for any measurable function in Lp for, p>0. This study investigated operators that a re good for best multiapproximation and best one sided multiapproximation. Approach: We first introduced some direct results related to the approximation problem of continuous functions by Hermit-Fejer interpolation based on the zeros of Chebyshev polynomials of the first or second kind i n terms of the usual modulus of continuity. They were then improved to spaces Lp for p 0.

On approximation of functions by trigonometric polynomials with incomplete spectrum in L p , 0 &lt; p &lt; 1

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L p , 0 < p <1, by trigonometric polynomials that are constructed by the system $$ \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} $$ . Bibliography: 13 titles.

Lp -moduli of continuity of sections of functions

Let ƒ(x, y) be a function in Lp(ℝ2) (1 ≤ p < ∞) and let ω1(ƒ; δ)p be the partial modulus of continuity of ƒ in Lp with respect to the variable x. For almost all y ∈ ℝ the sections ƒy(x) = ƒ(x, y) belong to Lp(ℝ). We find sharp estimates of the Lp-modulus of continuity of ƒy in terms of ω1(ƒ; δ)p.

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let Lp(G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual Lp space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space Lp(G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then Lp0(G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let Sp(G, H) be the L1(G) Banach module of functions in Lp(G, H) having unconditionally convergent Fourier series in Lp-norm. We show that Sp(G, H) coincides with the derived space Lp0(G, H), as in the scalar valued case. We also show that if G is compact and abelian, then Lp0(G, H) = L2(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ Lp(G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in Lp-norm, then F ∈ L2(G, H).

L p -boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ℒp, β + 1 (Γ) continuously onto \( \mathcal{N}_{p,\beta } (\Gamma ) \). The norm of a function in ℒp, β (Γ) is defined as Open image in new window , provided that the peak is at the origin. In this case, the norms on the spaces \( \mathcal{N}_{p,\beta }^ \mp (\Gamma ) \) are defined by Open image in new window , where q± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ℒp, β + 1 (Γ) onto ℳp, β(Γ), where ℳp, β(Γ) is the direct sum of \( \mathcal{N}_{p,\beta }^ + (\Gamma ) \) (Γ) and the space Open image in new window (Γ) of functions on Γ of the form p(z) = Σk = 0mt(k)Rezk with the parameter m = [μ − β − p−1]. The operator I − 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I − 2W continuously maps the space ℒp, β + 1 × ℒp, β + 1(Γ) to the space \( \mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma ) \).

INVERSE BORN APPROXIMATION FOR THE GENERALIZED NONLINEAR SCHRÖDINGER OPERATOR IN TWO DIMENSIONS

This work deals with the inverse scattering problems for the two-dimensional Schrödinger equation [Formula: see text] with a power-like nonlinearity, where the real-valued unknown functions αlon belong to [Formula: see text] with certain special behaviour at infinity. We prove Saito's formula which implies the uniqueness result and a representation formula for a sum of the functions αlin the sense of tempered distributions. What is more, we prove that the leading order singularities of this sum can be obtained exactly by the inverse Born approximation method from general scattering data at arbitrarily large energies. Especially, we show for the functions in Lp, for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces. In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by this scattering data.

L p approximation capability of RBF neural networks

Lp approximation capability of radial basis function (RBF) neural networks is investigated. If g: R+1 → R1 and \( g(\parallel x\parallel _{R^n } ) \) ∈ Llocp(Rn) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in Lp(K) with any accuracy for any compact set K in Rn, if and only if g(x) is not an even polynomial.

On moduli of smoothness and Fourier multipliers in L p , 0 &lt; p&lt; 1

We prove a theorem on the relationship between the modulus of smoothness and the best approximation in L p , 0 < p < 1, and theorems on the extension of functions with preservation of the modulus of smoothness in L p , 0 < p < 1. We also give a complete description of multipliers of periodic functions in the spaces L p , 0 < p < 1.

Inverse Born approximation for the nonlinear two-dimensional Schrödinger operator

This work deals with the inverse Born approximation for the nonlinear two-dimensional Schrödinger equation with cubic nonlinearity where the real-valued unknown functions q and α belong to with some behaviour at infinity. The following problem is studied: to estimate the smoothness of the terms from the Born sequence which corresponds to the scattering data with all arbitrary large energies and all angles in the scattering amplitude. These smoothness estimates allow us to conclude that the leading order singularities of the sum of unknown functions q and α can be obtained exactly by the Born approximation. Especially, we show for the functions in Lp, for certain values of p, that the approximation agrees with the true sum up to the functions from the Sobolev spaces. In particular, for the sum being the characteristic function of a smooth bounded domain this domain is uniquely determined by these scattering data.

A characterization of Lp(ℝ) by local trigonometric bases with 1&lt;p&lt;∞

We show that the local trigonometric bases introduced by Malvar, Coifman and Meyer constitute bases, but not unconditional bases, for Lp(ℝ) with 1<p<∞, p≠2. In addition, we characterize the functions in Lp(ℝ) for 1<p<∞ in terms of their local trigonometric basis coefficients.

Nearly monotone spline approximation in 𝕃_{𝕡}

It is shown that the rate of L p \mathbb {L}_p -approximation of a non-decreasing function in L p \mathbb {L}_p , 0 &gt; p &gt; ∞ 0&gt;p&gt;\infty , by “nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for “purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.

Universal approximation of symmetrizations by polarizations

Any symmetrization (Schwarz, Steiner, cap or increasing rear- rangement) can be approximated by a universal sequence of polarizations which converges in L p \mathrm {L}^p norm for any admissible function in L p \mathrm {L}^p for 1 ≤ p &gt; + ∞ 1 \le p &gt; +\infty and uniformly for admissible continuous functions. A new Pólya-Szegö inequality is proved for the increasing rearrangement.

Eigenvalues, Almost Periodic Functions, and the Derivative of an Integral

NOTES Edited by William Adkins Eigenvalues, Almost Periodic Functions, and the Derivative of an Integral Mark Finkelstein and Robert Whitley In [4], examples are given of functions g(x), bounded and continuous on (0, l], for which the indefinite integral g(t) dt is not differentiable from the right at zero. The purpose of this note is to show that such examples arise in a natural way and that the class of such examples is quite large. First, to see how such examples can come up, consider the operator l; Hf(x) = - I x 1x f (t) dt. A result due to Hardy [S, Equation 9.9.1 ], (7, Exercise 8.14] asserts that His a bounded linear operator on the Banach space LP[O, 1] for p > I with norm less than or equal to p/(p - 1). Actually Hardy gives the result for LP[O, oo), but it is somewhat simpler to work with LP[O, I]. Considering the function f(x) = x 0 (a real), which is in U[O, 1] when a > - 1/ p, computing Hf (x) , and letting a -4 - 1/ p shows that Hardy's bound for the norm of H is sharp and also shows that H is unbounded on L 1 (0, 1 ]. The case p = oo is a limiting case in which the norm of H is one. In working with the function f(x) = x 0 of the preceding paragraph, it will not have escaped the reader's notice that this function is an eigenfunction for H with eigenvalue 1/(1 +a). A basic operator theory question is: What are the eigenvalues of H? If A. is a nonzero eigenvalue with associated eigenvector f, then 11x f(t) dt x = A.f(x) , where equality holds almost everywhere. But equation (1) shows that f is absolutely continuous on (0, I], and ( 1) can be taken to hold not just almost everywhere but for all x in (0, l]. That is, a continuous f satisfying (l) for all x in (0, 1] can be chosen from the equivalence class of functions in LP[O, 1] that satisfy (1) almost everywhere. It is legitimate to differentiate equation (1) and, by rearranging, to get A.xj'(x) +(A. - l)f(x) = 0, which is an Euler differential equation [2, chap. 4, sec. 2] on (0, 1]. A priori (2) also holds only almost everywhere, but on solving for f' we see, as earlier, that a continuous representative from the equivalence class of f' can be chosen so that (2) holds for all x in (0, l]. It bas solutions of the form f(x) = xf3, where f3 =a+ ib, with f in LP [0, L] if a > -1 / p > - 1. The associated eigenvalue is A. = 1 / (1 + {3). It is immediate from (1) that the eigenfunction f is continuous at 0 (from the right, since f has domain [0, 1]) if and only if F (x) = f (t) dt is differentiable (from the right) at zero, and this occurs exactly when a > 0. l; August-September 2005] NOTES

Approximation of Functions in the Spaces L p (ℝ) by Means of Shifts and Contractions of One Function

Some approximation problems for functions of class Lp are considered. Bibliography: 4 titles.

Reconstruction of singularities in two dimensional Schrödinger operator with fixed energy

We prove that in dimension two potential scattering the leading order singularities of the unknown potential are obtained exactly by the Born approximation which corresponds to the scattering data with fixed energy. The proof is based on the new estimates for the Green-Faddeev's function in Lp(), 1 < p ≤ 2. These estimates allow us to consider the potentials with stronger singularities than in previous publications and with noncompact support.

N -widths on the classes of multivariate bandlimited functions *

Abstract In this paper, the tight order on the n-widths of the classes of multivariate bandlimited functions in Lp R i ), 1<p<∞ are determined.

Reconstruction of singularities in two dimensional Schrödinger operator with fixed energy

We prove that in dimension two potential scattering the leading order singularities of the unknown potential are obtained exactly by the Born approximation which corresponds to the scattering data with fixed energy. The proof is based on the new estimates for the Green-Faddeev's function in Lp(), 1 < p ≤ 2. These estimates allow us to consider the potentials with stronger singularities than in previous publications and with noncompact support.

The global approximation by Left-Bernstein-Durrmeyer quasi-interpolants inL p [0, 1

In this paper, we will use the 2r-th Ditzian-Totik modulus of smoothness ω 2rϕ (f,t) p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants M [2r−1]n f for functions of the space Lp[0,1] (1≤p≤+∞).

Lipoprotein (a) and atherosclerosis: a study from transgenic rabbits

High levels of lipoprotein (a) [Lp(a)] are an independent risk factor for a number of cardiovascular diseases such coronary heart disease, stroke and restenosis. In addition, numerous clinical and experimental studies indicate that Lp(a), similar to LDL, is an atherogenic lipoprotein, and thus, increased plasma concentrations of Lp(a) may be associated with premature atherosclerosis. Despite the progress in these studies during the past years, the precise physiological functions of Lp(a) is still unknown, and its pathological roles on the pathogenesis of atherosclerosis have not been fully defined. To investigate the functional roles of Lp(a), our laboratory has generated transgenic rabbits expressing human apolipoprotein (a) [apo(a)], and recently, we demonstrated that Lp(a) enhances diet-induced atherosclerosis and vascular calcification in transgenic rabbits. In this article, we summarize the progress in the study of human apo(a) transgenic rabbits.