Martingale transforms and uniformly convergent orthogonal series

S. A. Vinogradov's method is adapted to prove for certain orthogonal product systems the analogue of his inequality concerning the trigonometric system. For example, for the Walsh system W={w_n} the following holds. Let U(W) be the space of functions with a uniformly convergent Walsh-Fourier series. Then, for every functional F on U(W) we have the inequality .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 18–38, 1985.     

On the density of convergent subsequences of orthogonal series

A necessary and sufficient condition for (n) to be a Weyl factor for the existence, almost everywhere, of convergent subsequences of partial sums of an orthogonal series with upper density one is established.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 419–428, April, 1969.     

An explicit formula for the Fourier coefficients of Eisenstein series attached to lattices

We prove an explicit formula for local densities of inhomogeneous quadratic forms. The formula is derived by calculating explicitly the Fourier coefficients of various types of Eisenstein series.     

On a class of orthogonal series

, [0, 1], (n+1) n-. . [2]. — (. 5.4 5.6). . 6.4 2 [5]. , [4]. , , [6] [7]. [1].     

The absolute convergence of orthogonal series

We obtain sufficient conditions for the absolute convergence of Fourier series for functions of L _d ² depending on the properties of the function being expanded and the rate of growth of the sums of the system of functions {_k(t)} orthonormalized in [a, b] with respect to d(t). We show that if at some point x [a, b] the function (t) has a discontinuity, at that point the Fourier series of any functionf(t) L _d ² , converges absolutely.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 511–516, November, 1972.     

An explicit representation of Verblunsky coefficients

We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behavior, with remainder term, of those for FARIMA processes.     

The real field with convergent generalized power series

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on by a series with and for some is definable. This expansion is polynomially bounded.

     

The strongp-variation of martingales and orthogonal series

Let 1p< and letx=(x _n)n0 be a sequence of scalars. The strongp-variation ofx, denoted byW _p (x), is defined as

Weakly convergent sequences of functions and orthogonal polynomials

Measures and sequences of functions on locally compact spaces are studied, and a condition is given that, for a sequence of functions that is weakly convergent in L¹, ensures the strong convergence of a related sequence of functions. This result, together with a new integral formula for the reflection coefficients Φ_n(0) for the monic orthogonal polynomial Φ_n associated with a measure on the unit circle, is used to investigate convergence properties of orthogonal polynomials.     

Rearrangement of conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series _∑n∈ωan and every there is a permutation such that _∑n∈ωaπr(n)=r and

     

Absolutely convergent Fourier series and function classes. II

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes Λ_∗(α) and λ_∗(α) for some 0<α?2. This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168-1177] under the same title, and we keep its notations.     

On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients

A conjecture appears in Kumar and Sabanis (2016), in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficients. We answer this conjecture to the positive for the case of order 1.5 approximations and show that the suggested methodology works. Moreover, we explore the case of having Hölder continuous derivatives for the diffusion coefficients.     

Multiplier convergent series and uniform convergence of mapping series

In this paper, we introduce the frame property of complex sequence sets and study the uniform convergence of nonlinear mapping series in β-dual of spaces consisting of multiplier convergent series.     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

Coefficients of convergent multiple Haar series

It is well-known that if a multiple trigonometric series almost everywhere converges in the square or restricted rectangular sense to a finite function, then its coefficients grow slower than any exponential function. In this paper we prove the existence of a multiple Haar series that converges in the square or restricted rectangular sense to a finite function and contains a subsequence of coefficients that grows faster than any sequence defined in advance. Moreover, we show that for such series conditions of the Arutyunyan-Talalyan type can be violated at some points.