On the absolute summability of trigonometric series

{_n} a _{n n} , s _n n- , _n -n- (, 1)- . . . -, . , «|s₂n+1–s ₂ n¦< . .» «¦ _n – _n–1¦< . .» . s. , {a _n } 0, a _n/n<, {a _n } , {it|_n¦<. , , -.     

Universality and summability of trigonometric polynomials and trigonometric series

Periodica Mathematica Hungarica -     

The Lebesgue summability of trigonometric integrals

Motivated by the notion of Lebesgue summability of trigonometric series, we define the Lebesgue summability of trigonometric integrals in terms of the symmetric differentiability of the sum of the formally integrated trigonometric integral in question. We extend two theorems of Zygmund from trigonometric series to integrals, and one of them even in a more general form.     

On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.     

On the Lebesgue summability of regularly convergent double trigonometric series

We recall that the Lebesgue summability of the double trigonometric series (*) $$\sum\limits_{m \in \mathbb{Z}} {\sum\limits_{m \in \mathbb{Z}} {c_{m,n} e^{i(mx + ny)} } }$$ is defined in terms of the symmetric differentiability of its formally integrated series with respect to both variables. Under conditions weaker than the known ones in the literature, in this paper we prove that if the series (*) converges regularly at a point (x, y) to the sum s, then it is also Lebesgue summable at (x, y) to s (cf. the conditions (2.6) and ((2.7) in the known Theorem 1 and the conditions (3.1) and (3.2) in our new Theorem 2). This also demonstrates the superiority of the notion of regular convergence over the notion of convergence in Pringsheim’s sense of double series of numbers (see other examples in [5]).     

Strong summability and convergence of multiple trigonometric series over polyhedrons

The paper is a survey of the known results related with Fourier series strong polyhedral means of functions continuous on the m-dimensional torus, Sidon type inequalities for Dirichlet kernels in polyhedrons and integrability and convergence of multiple trigonometric series.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On totally regular summability methods

Mathematische Zeitschrift -     

Some theorems on the general summability methods

In this paper a new theorem which covers many methods of summability is proved. Several results are also deduced.