Orthonormal System Of Functions Research Articles

A New Approach to the Series Expansion of Iterated Stratonovich Stochastic Integrals with Respect to Components of a Multidimensional Wiener Process. The Case of Arbitrary Complete Orthonormal Systems in Hilbert Space

The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. This approach was proposed by the author in 2022 and is based on generalized multiple Fourier series in complete orthonormal systems of functions in Hilbert space. In the previous parts of this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 were obtained. At that, the expansions were constructed using two specific bases in Hilbert space. More precisely, Legendre polynomials and the trigonometric Fourier basis were used. In this paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 are obtained on the base of arbitrary complete orthonormal systems of functions in Hilbert space. Sufficient conditions for the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity are formulated in terms of trace series. The results of the article will be useful for construction of strong numerical methods with orders 1.0, 1.5 and 2.0 (based on the Taylor-Stratonovich expansion) for Ito stochastic differential equations with non-commutative noise.

Strichartz inequality for orthonormal functions associated with special Hermite operator

Abstract In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions.

On the Bounds for the Expected Maxima of Random Samples with Known Expected Maxima of Two Samples of Smaller Size

Our aim in the present paper is to give a new representation of the previously known estimates and further investigate upper and lower bounds for expected maxima of $n$ independent and identically distributed (i.i.d.) standardized random variables (r.v.'s) from known expected maxima of $m$ and $p$ r.v.'s with the same distribution, where $1<m<p<n$. A new representation is obtained from expansion of the inverse distribution function in a system of orthonormal functions on the unit interval. A criterion for attainability of the resulting bounds is put forward. We also obtain asymptotic properties of normed bounds for maxima expectations and normed maxima of r.v.'s with distribution where a criterion for attainability of these bound holds.

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Estimates of entropy for multiplier operators of systems of orthonormal functions

We obtain upper and lower estimates for ɛ -entropy and entropy numbers of multiplier operators of systems of orthonormal functions bounded from L p to L q . Upper estimates in our study require that a Marcinkiewicz-type multiplier theorem is available for the system. As application we obtain estimates for ɛ -entropy and entropy numbers of the multiplier operators associated with the sequences ( k − γ ln k − ξ ) k = 2 ∞ and ( e − γ k r ) k = 0 ∞ where γ > 0 , ξ ≥ 0 and 0 < r ≤ 1 . Some of these estimates are order sharp. We verify that the trigonometric system on the circle, the Vilenkin system and the Walsh system satisfy the conditions of our study. We also study analogous results for the Haar system and the Walsh systems on spheres.

Restriction theorem for the Fourier–Hermite transform and solution of the Hermite–Schrödinger equation

In this article, we prove a restriction theorem for the Fourier–Hermite transform and obtain a Strichartz estimate for the system of orthonormal functions for the Hermite operator $$H=-\Delta +|x|^2$$ on $${\mathbb {R}}^n$$ as application. Further, we show the optimal behavior of the constant in the Strichartz estimate as limit of a large number of functions.

Fractal Sobolev systems of functions associated with orthonormal systems of functions

This paper introduces the [Formula: see text]-fractal Sobolev system of functions corresponding to Sobolev orthonormal system of functions. An approximation-related result similar to Weierstrass theorem is derived. It is shown that the set of [Formula: see text]-fractal versions of Sobolev sums is dense and complete in the weighted Sobolev space [Formula: see text]. A Schauder basis and a Riesz basis of fractal type for the space [Formula: see text] are found. The Fourier–Sobolev expansion of an [Formula: see text]-fractal function [Formula: see text] corresponding to a certain set of interpolation points is presented. Moreover, some results on convergence of Fourier–Sobolev expansion of [Formula: see text] with respect to uniform norm and Sobolev norm are established.

Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs

We consider systems of functions (, ) that are Sobolev-orthonormal with respect to a scalar product of the form and are generated by a given orthonormal system of functions (). The Fourier series and sums with respect to the system (, ) are shown to be a convenient and efficient tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).

Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

We generalize the Lp spectral cluster bounds of Sogge for the Laplace–Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.

Finite Fourier decomposition of signals using generalized difference operator

In this paper, we introduce discrete inner product of two functions, discrete orthogonal and orthonormal system of functions and develop finite Fourier series for polynomial factorial, polynomial, exponential, rational and logarithm functions using the inverse of generalized difference operator.

Sidonicity and variants of Kaczmarz’s problem

We prove that a uniformly bounded system of orthonormal functions satisfying the ψ 2 condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher–Sidon property, and (3) must have its five-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the ψ 2 condition but which is not Sidon. These problems are variants of Kaczmarz’s Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new elementary proof of Pisier’s theorem that a set of characters satisfying the ψ 2 condition is Sidon.

Local smoothing for the quantum Liouville equation

We analyze in this work the regularity properties of the density operator solution to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schrödinger equation. We show in particular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.

On Fourier coefficients of lacunary systems

We prove that the Zygmund space L(lnL)1/2 is the largest among symmetric spaces X in which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients F(X) coincides with l2. Moreover, we obtain a description of spaces of Fourier coefficients corresponding to appropriate subsequences of arbitrary uniformly bounded orthonormal systems in symmetric spaces located between the spaces L(lnL)1/2 and L1.

Strichartz inequality for orthonormal functions

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

Kawahara solitons in Boussinesq equations using a robust Christov–Galerkin spectral method

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L2(-∞,∞) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives.As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

On properties of fourier series in general orthonormal systems of functions of bounded variation

On properties of fourier series in general orthonormal systems of functions of bounded variation

A positive density analogue of the Lieb–Thirring inequality

The Lieb-Thirring inequalities give a bound on the negative eigenvalues of a Schr\"odinger operator in terms of an $L^p$ norm of the potential. This is dual to a bound on the $H^1$-norms of a system of orthonormal functions. Here we extend these to analogous inequalities for perturbations of the Fermi sea of non-interacting particles, i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials.

Solving the equation for viscous incompressible liquid in a rectangular domain for small reynolds numbers

An algorithm is given for an analytical solution to the problem of the eigenvalue and eigenfunction for the Navier-Stokes equation within a rectangle under the adherence condition at its boundaries. The resulting system of orthonormal functions can be used for correctly deriving the divergence-free component of flow at the free surface of laboratory currents. This system makes it possible to apply the spectral approach to the solution of the Navier-Stokes equation, strictly calculate the primary regimes, investigate their stability, and calculate the secondary currents for small above-criticalities for the conditions of a laboratory experiment on simulating geophysical flows.

Solution of Population Balance Equations by the Finite Size Domain Complete Set of Trial Functions Method of Moments (FCMOM) for Inhomogeneous Systems

The FCMOM (finite size domain complete set of trial functions method of moments) is an efficient and accurate numerical technique to solve monovariate and bivariate population balance equations. It was previously formulated for homogeneous systems. In this paper, the FCMOM approach is extended to solve monovariate population balance equations for inhomogeneous (spatially not uniform) systems. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle size distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. The FCMOM is extended to inhomogeneous systems assuming that the particle-phase convective velocity is independent of the internal variables (particle size). The method is illustrated by applications to particle diffusion and to particle convection. In the case of particle convection, a gas−solid dilute flow in a pipe was simulated and the FCMOM equations were coupled with the governing...

Transferring strong boundedness among Laguerre orthogonal systems

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted Lp of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted Lp boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials.