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It is shown that a locally compact groupoid with open range map does not always admit a Haar system. It then is shown how to construct a Haar system if the stability groupoid and the quotient by the stability groupoid both admit one. [ABSTRACT FROM AUTHOR]

In this paper we discuss the uniqueness property of a summation method for multiple series with respect to Vilenkin and generalized Haar systems. It is proved that if the multiple series with respect to these systems is a.e. summable by that method to an integrable function on [0; 1)d and satisfies an extra condition, then it is the Fourier series of this function. [ABSTRACT FROM AUTHOR]

We start from the question raised by Williams (Proc. Am. Math. Soc. 2016): Must a second countable, locally compact, transitive groupoid G have open range map? If the answer is positive, the topology of G is in fact locally transitive (in the sense of [Seda, 1976]). We prove that even if the answer is negative, we can replace the original topology of G with a local transitive topology so that the topologies of the r-fibres are not affected. The new topology is locally compact Hausdorff but not necessary second countable. However its full C*-algebra (introduced in [Renault, 1980]) is still isomorphic to C*(H)⊗K(L2(μ)), where H is the isotropy group at a unit u and μ is a positive Radon measure on the unit space. We also present a few remarks concerning the Haar systems on locally compact groupoids and for every locally compact groupoid having paracompact unit space and second countable r-fibres, we prove the existence of a pre-Haar system bounded on the compact sets. [ABSTRACT FROM AUTHOR]

In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev-Triebel-Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel-Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters. [ABSTRACT FROM AUTHOR]

We consider the classical ill-posed problem of recovering of a function from noisy Fourier coefficients with respect to the Haar system. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound. [ABSTRACT FROM PUBLISHER]

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 second countable locally compact Hausdorff groupoids, the property of possessing a Haar system is preserved by equivalence. [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.

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 class of functions of s variables is defined. For the functions of this class, Sász's theorem for s-dimensional Fourier-Haar series is proved. When , it is shown that the obtained theorem is best possible for general complete orthonormal systems. [ABSTRACT FROM AUTHOR]

W.D. Munn proved that a finite-dimensional representation of an inverse semigroup is equivalent to a ∗-representation (by partial isometries) if and only if it is bounded. This paper gives a new analytic proof that every finite-dimensional representation of a compact inverse semigroup is equivalent to a ∗-representation. This will be the main result of this paper. [ABSTRACT FROM AUTHOR]

We obtain an exact description for the behavior of Fourier-Haar sums at dyadic irrational discontinuity points for functions of bounded variation. [ABSTRACT FROM AUTHOR]

For a wide class of Young functions … :[0, ∞) ➔ [0, ∞) we determine the best constant C… such that the following holds. If(hk)k≥0 is the Haar system on[0,1] then for any vectors ak from a separable Hilbert space H and …k∈{-1,1}, k=0,1,2,…, we have … This is generalized to the sharp weak-… inequality … where X, Y stand for H-valued martingales such that Y is differentially subordinate to X. These statements complement and generalize the results of Burkholder, Suh, the author and others. [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 calculate the norms of multipliers for the Haar system in some rearrangement invariant spaces for which the Haar system is not an absolute basis. [ABSTRACT FROM AUTHOR]

Abstract: The -discrepancy measures the irregularity of the distribution of a finite point set. In this note, we prove lower bounds for the -discrepancy of arbitrary -point sets. Our main focus is on the two-dimensional case. Asymptotic upper and lower estimates of the -discrepancy in dimension 2 are well known, and are of the sharp order . Nevertheless, the gap in the constants between the best-known lower and upper bounds is unsatisfactorily large for a two-dimensional problem. Our lower bound improves upon this situation considerably. The main method is an adaption of Roth’s method, using the Fourier coefficients of the discrepancy function with respect to the Haar basis. We obtain the same improvement in the quotient of lower and upper bounds in the general -dimensional case. Our lower bounds are also valid for the weighted discrepancy. [Copyright &y& Elsevier]

Abstract: For any we construct a Weak Thresholding Greedy Algorithm with weakness parameter which converges in with respect to the Haar system, i. e. the Haar system is a ‘good non quasi-greedy basis’. [Copyright &y& Elsevier]

Abstract: If X is a compact convex set in a real locally convex space, is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set. [Copyright &y& Elsevier]

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 Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) spaceEif and only if the multiplier determined by the sequenceforn= 0 andk= 0, 1, . . . , 2n forn>1, is bounded inE. If the Lorentz space ?(?) differs fromL1 andL8 then there is a multiplier with respect to the Haar system which is bounded in ? (?) and unbounded inL8 andL1. [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.