General Orthogonal System Research Articles

Dense weakly lacunary subsystems of orthogonal systems and maximal partial sum operator

It is shown that any finite orthogonal system of functions whose norms in $L_p$ are bounded by 1, where $p>2$, has a sufficiently dense subsystem with lacunarity property in the Orlicz space. The norm of the maximal partial sum operator for this subsystem has a better estimate than it is guaranteed by the classical Menshov-Rademacher theorem for general orthogonal systems. Bibliography: 17 titles.

On Convergence Properties of Tensor Products of Some Operator Sequences

We consider sequences of compact bounded linear operators \(U_n:L^p(0,1)\rightarrow ~L^p(0,1)\) with certain convergence properties. Several divergence theorems for multiple sequences of tensor products of these operators are proved. These theorems in particular imply that \(L\log ^{d-1} L\) is the optimal Orlicz space guaranteeing almost everywhere summability of rectangular partial sums of multiple Fourier series in general orthogonal systems.

Enabling quaternion derivatives: the generalized HR calculus.

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

Magnetohydrodynamic equilibrium. II - General integrals of the equations with one ignorable coordinate

The steady equations of hydromagnetics for the isentropic or nonisentropic flow of an inviscid magnetofluid of high electrical conductivity, with one ignorable coordinate in a general orthogonal system, are treated. Several integrals of the equations are established thereafter reducing them to a scalar, quasi-linear, second order, partial differential equation for the magnetic potential. Simple solutions of this final equation are presented. The result, together with a similar treatment of helically symmetric hydromagnetic flows presented in a subsequent paper, allows a unified and systematic approach to the solution of problems involving steady hydromagnetic fields with a topological invariance in various curvilinear coordinates.

Book Review: Fourier series with respect to general orthogonal systems

Book Review: Fourier series with respect to general orthogonal systems