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This article, titled "Functions Almost Universal in the Sense of Signs with Respect to the Trigonometric System and the Walsh System," discusses the existence and structure of functions whose Fourier series are almost universal in the sense of signs with respect to the trigonometric system and the Walsh system. The paper explores various definitions and theorems related to universal functions and poses unanswered questions about the existence of certain types of universal functions. The authors provide examples and results from previous studies in the field of complex and real analysis. The article concludes with acknowledgments and funding information. [Extracted from the article]

According to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. We extend Wik's result to the multidimensional case. [ABSTRACT FROM AUTHOR]

In this paper, the problem of -convergence of Fourier series with quasi-monotone coefficients is handled by using the -mean. Also, an example is given about the Fourier series of a signal (function) and its mean. [ABSTRACT FROM AUTHOR]

We study the convergence of Fourier series in the polynomial system orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of in this system converges to pointwise on the grid as . In addition, we study the approximation properties of partial sums of Fourier series in the system . [ABSTRACT FROM AUTHOR]

We obtain an estimate for the convergence rate of the Fourier series of a continuous periodic function in terms of the modulus of continuity of the function and the value of its -variation. We prove that the leading term of the estimate is sharp. [ABSTRACT FROM AUTHOR]