Differentiation with respect to nets and the Haar series

We set out the connection between convergence of the Haar series and differentiation with respect to nets of a function. This connection allows us to give a new proof of certain earlier theorems on Haar series, and also to prove a number of new generalizations.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 33–40, July, 1968.In conclusion we remark that the results we have proved here for the Haar series can be extended to the series in Haar-type systems considered at the end of [4].     

On the multipliers of Fourier series with respect to the Haar system

qV РАБОтЕ РАссМАтРИВАУ тсь МУльтИплИкАтИВН ыЕ пРЕОБРАжОВАНИь РьДО В ФУРьЕ-хААРА. ИжУЧЕНы слЕДУУЩИЕ кл Ассы МУльтИплИкАтОР ОВ: (H _p ^α ,H _q ^β ), (H _p ^α ,L ^q ) И (с, с).     

Convergence of fourier series with respect to multiplicative systems and the p-fluctuation continuity modulus

We obtain some sufficient conditions for convergence of series of the Fourier coefficients with respect to multiplicative systems for functions of bounded p-fluctuation. In some cases we establish the unimprovability of these conditions.     

The Parseval equality for multiple Fourier-Stieltjes series with respect to the Haar system

Let a function f : \(\Pi ^{ * ^m } \) → ? be Lebesgue integrable on \(\Pi ^{ * ^m } \) and Riemann-Stieltjes integrable with respect to a function G : \(\Pi ^{ * ^m } \) → ? on \(\Pi ^{ * ^m } \). Then the Parseval equality

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holds, where

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(k) = (f, χk) = (L)

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f(x)χk(x) dx and \(\widehat{dG}\)(k) =

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χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Π ^m → ? is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Π ^m ,

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are Fourier-Lebesgue coefficients of the function f _x(t) = f(x ⊕ t), where ⊕ is the group addition, then the Parseval equality

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holds for almost all x ∈ \(\Pi ^{ * ^m } \) in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).

     

Uniqueness of series with respect to general Franklin systems

We give some sufficient condition for the uniqueness of series with respect to general Franklin systems and discuss analogous results for orthonormal spline systems of higher order and some wavelet systems.     

Convergence of fourier series with respect to an orthogonal system of functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 641–647, May, 1989.     

On the uniqueness of a Haar series that converges with respect to subsequences of partial sums

We prove the existence of two different series in the Haar system for which the subsequences of partial sums converge to the same D-integrable function everywhere on [0, 1].Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 707–714, December, 1968.     

Absolute convergence of Fourier series with respect to bounded systems

In this paper we prove that for the piecewise-linear unit jump approximation the sum of the moduli of the Fourier coefficients with respect to an arbitrary complete orthonormal system, which is totally bounded, has, when averaged over sections, a lower bound of order log N, where N^?1 is the approximation step.     

Summability of series with respect to a Haar system by the (C, 1) method

For a Haar-system series we prove that if the lower bound of the (C, 1) means of the series is larger than — ∞ on a set E of positive measure, then the series converges to a finite function almost everywhere on E; from this it follows that Haar-system series are not summable by the (C, 1) method to + ∞ on sets of positive measure.     

Divergence of the Fourier series by generalized Haar systems at points of continuity of a function

We obtain a connection between the Dirichlet kernels and partial Fourier sums by generalized Haar and Walsh (Price) systems. Based on this, we establish an interrelation between convergence of the Fourier series by generalized Haar and Walsh (Price) systems. For any unbounded sequence we construct a model of continuous function on a group (and even on a segment [0, 1]), whose Fourier series by generalized Haar system generated by this sequence, diverges at some point.