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1.
Ferenc Móricz 《Journal of Fourier Analysis and Applications》2001,7(6):601-614
A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth
partial sum of its formally differentiated Fourier series divided by n converges to π-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation
is also well known.
The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes
series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause.
As corollaries, we also obtain the following results:
相似文献
(i) | The terms of the Fourier or Fourier-Stieltjes series of F(x, y) determine the atoms of the (periodic) Borel measure induced by (an appropriate extension of) F. |
(ii) | In the case of periodic functions F(x, y) of bounded variation, the class of double Fourier-Stieltjes series coincides with the class of series that can be obtained from their Fourier series by a formal termwise differentiation with respect to both x and y. |
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The trigonometric polynomials of Fejér and Young are defined by $S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}}
{k}}$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}}
{k}} and $C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}}
{k}}$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}}
{k}}, respectively. We prove that the inequality $\left( {{1 \mathord{\left/
{\vphantom {1 9}} \right.
\kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/
{\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right.
\kern-\nulldelimiterspace} {S_n \left( x \right)}}$\left( {{1 \mathord{\left/
{\vphantom {1 9}} \right.
\kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/
{\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right.
\kern-\nulldelimiterspace} {S_n \left( x \right)}} holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp. 相似文献
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《复变函数与椭圆型方程》2012,57(7):589-598
Let m and n denote a pair of positive integers. In this paper, we call upon the Hadamard product and computer algebra techniques to evaluate the Fejér integral π ?1 ∫0 π (sin mθ / sin θ) 2n dθ. Using symmetry arguments, it is proved that the value of this integral is an odd polynomial in m of degree 2n ? 1. This permits using polynomial curve fitting methods and mathematical software packages to obtain evaluation formulas for n relatively small. Some cases of the above integral with 2n replaced by 2n + 1 are also discussed. A familiar identity shows that these yield evaluations of integrals of powers of certain Tchebychev polynomials. 相似文献
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In this paper we prove that there are operators in the uniform Roe algebra ${C_u^*(G)}$ which cannot be approximated by the truncations of themselves, where G is a finitely generated group. We also give a sufficient condition for the operators which can be approximated by the truncations of themselves. For a countable discrete metric space X, we obtain the similar conclusions. 相似文献
9.
Получены асимптотич еские равенства для в еличин гдеr≧0 — целое, ω(t) — выпу клый модуль непрерыв ности и $$\bar \sigma _n (f;x) = - \frac{1}{\pi } \mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2}ctg\frac{t}{2} - \frac{1}{{4(n + 1)}}\frac{{\sin (n + 1)t}}{{\sin ^2 \tfrac{1}{2}t}}} \right)dt$$ сумма Фейера функцииf(х), сопряженной сf(x). 相似文献
10.
M. Avdispahic 《Rendiconti del Circolo Matematico di Palermo》1986,35(1):90-101
La successione di elementi di una serie di Fourier derivata di una funzione appartenente alla classe di WienerV p,p>1 (rispettivamente alla classe di Waterman {n β}B V o à quella di ChanturiyaV [n β] per qualsiasi 0<β<1) è sommabile verso (f(x+0)?f(x?0))/π mediante i metodi di Cèsaro di ordine α>1?1/p (α>β). 相似文献
11.
It is proved that the operators σ n Δ of the triangular-Fejér-means of a two-dimensional Walsh-Fourier series are uniformly bounded from the dyadic Hardy space H p to L p for all 4/5 < p≤∞. 相似文献
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Ferenc Weisz 《Analysis Mathematica》2002,28(2):135-155
We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space H
p([0,1)d
1 × ¨ × [0,1)d
l) to L
p([0,1)d) if 1/2 < p < ∞ and m
j ≥ 0, |k
j| ≤ m
j + 1. By an interpolation theorem, we get that the maximal operator is also of weak type (H
1
#i,L
1) (I=1,¨,l), where the Hardy space H
1
#i is defined by a hybrid maximal function and H
1
#i
L(logL)l-1. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f H
1
#i
and converge in a cone if f ∈ L
1. 相似文献
14.
V. V. Savchuk 《Mathematical Notes》2007,81(5-6):665-670
For the Dirichlet classes D p of holomorphic functions in the disk, we obtain the exact orders of best polynomial approximations and of upper bounds for deviations of Fejér means of Taylor series in the Hardy spaces H p. 相似文献
15.
The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003 相似文献
16.
The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with π. 相似文献
17.
U. Goginava 《Acta Mathematica Hungarica》2007,115(4):333-340
The main aim of this paper is to prove that the maximal operator of the Fejér mean of the double Walsh-Fourier series is not bounded from the Hardy space H 1/2 to the space weak-L 1/2. This paper was written during the visit of the author at the College of Nyíregyháza in Hungary. 相似文献
18.
The paper deals with the S-technology, which reduces convex problems of quadratic programming to the solution of systems of several linear, and one convex, inequalities. A certain variant of the Fejér method is applied to these systems. In particular, the problem of the constructive separability of convex polyhedral sets by a layer of maximal thickness is solved. This algorithm plays an important role in problems of discriminant analysis. 相似文献
19.
V. V. Vasin 《Russian Mathematics (Iz VUZ)》2009,53(2):1-20
In the latter thirty years, the solution of ill-posed problems with a priori information formed a separate field of research in the theory of ill-posed problems. We mean the class of problems, where along with the basic equation one has some additional data on the desired solution. Namely, one states some relations and constraints which contain important information on the object under consideration. As a rule, taking into account these data in a solution algorithm, one can essentially increase its accuracy for solving ill-posed (unstable) problems. It is especially important in the solution of applied problems in the case when a solution is not unique, because this approach allows one to choose a solution that meets the reality. In this paper we survey the methods for solving such problems. We briefly describe all relevant approaches (known to us), but we pay the main attention to the method proposed by us. This technique is based on the application of iterative processes of Fejér type which admit a flexible and effective realization for a wide class of a priori constraints. 相似文献
20.
The interpolating spline or trigonometric polynomial to a function at equally spaced points approximates the Dirichlet partial sums of its Fourier series with accuracy depending only on the neglected coefficients. We show that the Fejér mean of the Dirichlet sums can be approximated by the arithmetic mean of two Fejér trigonometric interpolants, one at the points with even indexes and one at the points with odd indexes, with an error depending only on the neglected Fourier coefficients and it is positive for positive functions. We also consider the case of Fejér spline interpolants and a constructive relation between Hermite and Fejér interpolants. 相似文献