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A uniqueness theorem and a recovery theorem for the coefficients of series in generalized Haar systems are proved under the assumption that the series converge in measure and satisfy a certain necessary condition on the distribution function of the majorant of partial sums. [ABSTRACT FROM AUTHOR]

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classicalDini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained. [ABSTRACT FROM AUTHOR]

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson's inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables. [ABSTRACT FROM AUTHOR]

For weighted symmetric (or rearrangement-invariant) spaces with nontrivial Boyd indices and weights from suitable Muckenhoupt classes, the basis property of the Haar system in these spaces and two versions of the direct theorem on the approximation by polynomials in the Haar system are established. [ABSTRACT FROM AUTHOR]

The properties of the Fourier coefficients for some classes of functions with respect to both the Haar system and general orthonormal systems are studied. It is established that the results obtained in the paper cannot be strengthened. [ABSTRACT FROM AUTHOR]

The article focuses on the convergence of Fourier series with respect to general orthonormal systems. Topics discussed include Lebesgue function of the orthonormal system; implying the existence of a function through Banach-Steinhaus theorem; and Walsh system, or a Haar system for the implying the equivalence classes.

The article focuses on development of uniqueness theorems involved in the Franklin system. Topics discussed include evolution of first Orthonormal basis by Ph. Franklin, establishment of uniqueness theory for trigonometric series by use of Cantor uniqueness theorem, achievement of mathematical proofs for Haar series by Contor theorem, limited studies on problems related to franklin system series and several theorems are mentioned of Fourier-Franklin series for estimation of Dirichlet kernel.

In this paper, best canonical n-term approximations in the norm of the spaces L(0, 1) of the family I of characteristic functions of intervals are studied. [ABSTRACT FROM AUTHOR]

The article focuses on a study which explored the resemblance between different Haar bases. It mentions the general scheme described by the study for the development of Haar bases that does not depend on the group operation +. Moreover, the study showed that all Haar bases can be determined by similar formulas.

We present an algorithm for constructing dilation operators on the product of groups of p-adic integers and construct a system of Haar functions which is obtained from a single function by using the operations of contraction, translation, and raising to a power. In the two-dimensional case, we describe all the Haar bases. [ABSTRACT FROM AUTHOR]

It is well known that if the Fourier-Haar coefficients have a certain order or if a certain series composed of the Fourier-Haar coefficients of a function f( x) ∈ C(0, 1) converges, then the function has a certain form. In the present paper, we prove that not only the Fourier-Haar coefficients, but also the difference of these coefficients possess these properties. [ABSTRACT FROM AUTHOR]

We prove that du Bois-Reymond-type theorems do not hold for the double Haar series, for the dyadic cubic Henstock integral, and for regular convergence over rectangles. [ABSTRACT FROM AUTHOR]

We obtain conditions for the convergence in the spaces L p [0, 1], 1 ≤ p < ∞, of biorthogonal series of the form in the system { ϕ n } n≥0 of contractions and translations of a function ϕ. The proposed conditions are stated with regard to the fact that the functions belong to the space $$ \mathfrak{L}^p $$ of absolutely bundleconvergent Fourier-Haar series with norm where ( f,χ n ), n = 0, 1, ..., are the Fourier coefficients of a function f ∊ L p [0, 1] in the Haar system {χ n } n≥0. In particular, we present conditions for the system { ϕ n } n≥0 of contractions and translations of a function ϕ to be a basis for the spaces L p [0, 1] and $$ \mathfrak{L}^p $$ . [ABSTRACT FROM AUTHOR]

In this paper, we study a special perturbation of the function system obtained from the Fejér kernel. It is shown how this relates to the stability of bases and complete systems as well to the stability of the trigonometric system. An approximation algorithm in systems resulting from a perturbation of the original system is given. [ABSTRACT FROM AUTHOR]

We study the universality of a Dirichlet series with periodic coefficients. This property is proved in the case of multiplicative coefficients, and in the general case we establish universality in a certain set of analytic functions related to a probability distribution. [ABSTRACT FROM AUTHOR]

In this paper, we study whether the set A(ϕ) is closed under multiplication f · g, where f and g belong to the class A(ϕ). We also study the problem of the existence of a solution of the equation Bx = C (where B,C ∈ A(ϕ) and B ≠ 0) on the set A(ϕ). [ABSTRACT FROM AUTHOR]

The article considers the m-term L¹-greedy algorithm in the Haar system normalized in L¹ (0,1). It uses the binary numbering of the functions of the Haar system. It describes some notation and proves several lemmas, which will be used in the proof of the theorem. It proves the existence of a function f ε L¹ (0,1) for which there is no L¹-greedy algorithm in the Haar system converging to f.

The main result of this paper is theorem asserting that for a series in the generalized Haar system generated by bounded functions, all of its subseries (and even a wider class of series) are Fourier series in the sense of the A-integral of its sum (although they are not necessarily standard Fourier series). [ABSTRACT FROM AUTHOR]

The article focuses on the deviation of partial sums of Fourier-Haar systems with arbitrary dilation factor. It states that Fourier series converges uniformly for any continuous 1-periodic function. It adds that at the points of discontinuity, the values of the functions are taken as the half-sum of the left and right limits.

The article presents the Haar system rearrangements in the binary spaces of H and BMO, where BMO is the binary space dual to H. The theorem which refines and generalizes Schipp's theorem is presented. The necessary and sufficient conditions for the boundedness of the operators is established by Semenov's theorem.

The article presents the theorems of the Haar system.