Class Lip Research Articles

Degree of approximation of functions conjugate to the functions belonging to Lip((𝜉1,𝜉2)𝑟)-class through double Nörlund means

Abstract In this paper, we determine the error of approximation of functions, conjugate to the functions of two variables ( 2 ⁢ π 2\pi -periodic in both variables) belonging to the Lipschitz class Lip ( ( ξ 1 , ξ 2 ) ; r \operatorname{Lip}((\xi_{1},\xi_{2});r ) ( r ≥ 1 r\geq 1 ), through the double Nörlund means of its conjugate double Fourier series. Additionally, in the form of corollaries, we present some particular cases of the theorem proved here.

On Paracompactness of the Space CLipk(Ω)

By the construction C^Lipk(Ω) we mean the space of all continuous complex-valued functions of class Lip^k, defined on an open subset Ω of Euclidean space ℝⁿ the present work, proved the paracompactness of the space C^Lipk(Ω), also many important results were discussed and proved.

Trigonometric approximation of functions f(x,y) of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y) belonging to the generalized Lipschitz class Lip ((xi _{1}, xi _{2} );r ), r geq 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip ((alpha ,beta );r ) and Lip(alpha ,beta ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C, gamma , delta ) means.

Human motor cortex relies on sparse and action-specific activation during laughing, smiling and speech production

Smiling, laughing, and overt speech production are fundamental to human everyday communication. However, little is known about how the human brain achieves the highly accurate and differentiated control of such orofacial movement during natural conditions. Here, we utilized the high spatiotemporal resolution of subdural recordings to elucidate how human motor cortex is functionally engaged during control of real-life orofacial motor behaviour. For each investigated movement class—lip licking, speech production, laughing and smiling—our findings reveal a characteristic brain activity pattern within the mouth motor cortex with both spatial segregation and overlap between classes. Our findings thus show that motor cortex relies on sparse and action-specific activation during real-life orofacial behaviour, apparently organized in distinct but overlapping subareas that control different types of natural orofacial movements.

Degree of Approximation by a General Cλ-Summability Method

In the present study, two theorems explaining the degree of approximation of signals belonging to the class Lip(α, p, w) by a more general Cλ-method (Summability method) have been formulated. Improved estimations have been observed in terms of λ(n); where (λ(n))−α ≤ n−α for 0 < α ≤ 1 as compared to previous studies presented in terms of n. These estimations of infinite matrices are very much applicable in solid state physics which further motivates for an investigation of perturbations of matrix valued functions.

Approximation of signals (functions) of Lip(α, p), (p &gt; 1)-class by trigonometric polynomials

Given a function f in the class Lip(α,p) (0 < α ≤ 1, p > 1), Mittal and Singh (2014) approximated such an f by using trigonometric polynomials, which are the nth terms of either certain Riesz mean or Nцrlund mean transforms of the Fourier series representation for f. They showed that the degree of approximation is O((λ(n))−α) and extended two theorems of Leindler (2005) where he had weakened the conditions on {pn} given by Chandra (2002) to more general classes of triangular matrix methods. We obtain the same degree of approximation for a more general class of lower triangular matrices.

On the approximation by Chlodowsky type generalization of (p,q)-Bernstein operators

In the present article, we introduce Chlodowsky variant of $(p,q)$-Bernstein operators and compute the moments for these operators which are used in proving our main results. Further, we study some approximation properties of these new operators, which include the rate of convergence using usual modulus of continuity and also the rate of convergence when the function $f$ belongs to the class Lip$_{M}(alpha )$. Moreover, we also discuss convergence and rate of approximation in weighted spaces and weighted statistical approximation properties of the sequence of positive linear operators defined by us.

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip k

In the present paper we prove a criterion of Lip k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Frechet space F. We establish that a manifold is Lip k -paracompact if and only if the model space F is paracompact and Lip k -normal. We prove a sufficient condition for existence of Lip k -partition of a unity on a manifold of class Lip k .

Convergence of the Fourier Series of Lip 1 Functions with Respect to General Orthonormal Systems

We establish sufficient conditions that should be satisfied by functions of a general orthonormal system (ONS) {φ n (x)} in order that the Fourier series in this system be convergent almost everywhere on [0, 1] for any function from the class Lip 1. It is shown that the obtained conditions are best possible in a certain sense.

Regularity of the Beurling Transform in Smooth Domains

UDC 517.5 The relationship between smoothness properties of the boundary of a domain Ω and the boundedness of the Beurling transform in the corresponding Lipschitz classes Lip(ω) for the case of a Diniregular module of continuity ω is studied. The result is sharp. Our motivation arises from the work of Mateu, Orobitg, and Verdera. Bibliography: 8 titles.

Summability of general orthonormal Fourier series

S. Banach in [1] proved that for any function f ∈ L2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α &gt; 0. D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α &gt; 0. In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φn(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

Approximation of a Function Belonging to the Class Lip (ψ (t), p) by Using [s,a&lt;sub&gt;n&lt;/sub&gt;] Means

Approximation of a Function Belonging to the Class Lip (ψ (t), p) by Using [s,a&lt;sub&gt;n&lt;/sub&gt;] Means

Trigonometric approximation of functions belonging to certain Lipschitz classes by C1⋅ T operator

The study of approximation properties of the periodic functions in Lp(p ≥ 1)-spaces, in general and in Lipschitz classes Lip α, Lip (α, p), Lip (ξ(t), p) and weighted Lipschitz class W(Lp, ω(t), β), in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals [E. Z. Psarakis and G. V. Moustakides, An L2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Systems I Fundam. Theory Appl., 44(7) (1997) 551–601]. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x) - Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f. In this paper, we discuss the approximation properties of the periodic functions in the Lipschitz classes Lip α and W(Lp, ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix means of the Fourier series associated with the function. The degree of approximation obtained in our theorems of this paper is free from p and sharper than earlier results.

Wavelet expansions on the Cantor group

Wavelet expansions in Lp-spaces on the locally compact Cantor group G are studied. An order-sharp estimate of the wavelet approximation of an arbitrary function f ∈ Lp(G) for 1 ≤ p < ∞ in terms of the modulus of continuity of this function is obtained, and a Jackson-Bernstein type theorem on approximation by wavelets of functions from the class Lip(p)(α; G) is proved.

Error estimates of cubature formulas exact for Haar polynomials in classes of two-variable functions satisfying a general Lipschitz condition

Upper error estimates are obtained for cubature formulas with the Haar d-property in the classes Lip(L1, L2) of two-variable functions satisfying a general Lipschitz condition. It is shown that the error of minimal cubature formulas possessing the Haar d-property have the best order of convergence to zero in the indicated classes.

Sufficient conditions for the Lebesgue integrability of double Fourier transforms

We consider complex-valued functions \(\widehat{f}\) for some 1 > p = 2 and give sufficient conditions for its Fourier transform f to belong to \(\widehat{f}\epsilon L^{r}(\mathbb{R}^2)\), where 0 > r > q and 1/p + 1/q = 1. Under additional conditions, we also give sufficient conditions, under which we have f Lr(R2). These sufficient conditions are in terms of the Lp-integral modulus or the ordinary modulus of continuity of f. Our theorems apply for functions in the Lipschitz classes Lip(α{in1}, α{in2}), where 0 > α{in1}, α{in2} ≤ 1 as well as for functions of bounded {its}-variation on R2, where 0 > {its} > {itp}. The results of this paper can be considered to be the nonperiodic versions of those results proved in [5] for double Fourier series, and the latter ones were in turn the two-dimensional extensions of the classical theorems of Bernstein, Szász and Zygmund on the absolute convergence of single Fourier series.

Euler-Hausdorff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class

In this paper, three new estimates for the degree of approximation of a function ˜ f ,t he conjugate of a function f belonging to classes Lip α and Lip(ξ , r), r ≥ 1, by E (q) · � H summability operator of conjugate series of the Fourier series have been determined. MSC: 42A24; 41A25; 42B05; 42B08

Smoothness of functions in the metric spaces L ψ

Let L 0 (T ) be the set of real-valued periodic measurable functions, let ψ : R + → R + be a modulus of continuity (ψ ≠ 0) , and let $$ {L_{\uppsi }}\equiv {L_{\uppsi }}(T)=\left\{ {f\in {L_0}(T):{{{\left\| f \right\|}}_{\uppsi }}:=\int\limits_T {\uppsi \left( {\left| {f(x)} \right|} \right)dx<\infty } } \right\}. $$ The following problems are investigated: the relationship between the rate of approximation of f by trigonometric polynomials in L ψ and the smoothness in L 1, the relationship between the moduli of continuity of f in L ψ and L 1 and the imbedding theorems for the classes Lip(α, ψ) in L 1, and the structure of functions from the class Lip(1, ψ).

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip α. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.

TRIGONOMETRIC APPROXIMATION BY MATRIX TRANSFORMS IN Lp(x)SPACES

The Lipschitz classes Lip (α, p(x)), 0 &lt; α ≤ 1 are defined for the Lebesgue spaces Lp(x)with variable exponent p(x), and the degree of approximation by matrix transforms of f ∈ Lip (α, p(x)) is estimated by n-α.