Search Results

You are looking at 1 - 8 of 8 items for :

  • "trigonometric system" x
Clear All

Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f  we take as an approximant a trigonometric polynomial of the form  G m(f ) := ?k?? f^(k) e (i k,x), where ??Zd  is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that G m(f ) gives the best m-term approximant in the L 2-norm and, therefore, for each f ?L 2,  ¦f-G m(f 2?0  as  m ?8.  It is known from previous results that in the case of p ?2 the condition f ?L p does not guarantee the convergence  ¦f-G m(f p?0  as  m ?8..  We study the following question. What conditions (in addition to f ?L p) provide the convergence  ¦f-G m(f p?0  as  m ?8?  In our previous paper [10] in the case 2< p =8 we have found necessary and sufficient conditions on a decreasing sequence {A n} n =1 8 to guarantee the L p-convergence of {G m(f )} for all f ?L p   , satisfying a n (f ) =A n , where {a n (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ?L p and  ¦G M (m)(f ) - G m(f p  ?0  as  m ?8 imply  ¦fG m(f p  ?0  as  m ?8. We have found these conditions in the case when p is an even number or p = 8.

Restricted access

Abstract  

The behavior of generalized Cesàro (C, α n)-means (α n ∈ (0, d), d > 0) of trigonometric Fourier series of H ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.

Restricted access

Abstract  

The behavior of generalized Cesro (C, α n)-means (α n ∈ (−1,0)) of trigonometric Fourier series of H ω classes in the space of continuous functions is studied. The sharpness of the results obtained is shown.

Restricted access

Abstract  

The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L p and C-modulus of continuity is studied.

Restricted access

Abstract  

The behaviour of the Cesàro means of trigonometric Fourier series of monotone type functions in the space of continuous functions is studied.

Restricted access

Abstract

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro (C,α n)-means (α n∊(−1,0)) of trigonometric Fourier series of the classes H ω∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown.

Restricted access

Abstract  

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function fL 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

Restricted access

Abstract  

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S n(·, f)-f(·)‖C [0,2π] where S n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained.

Restricted access