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Ilona Iglewska-Nowak

Publications and source records attributed to Ilona Iglewska-Nowak.

17 recordsLinked to original sources
arXiv2025

Wavelet based solutions to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere

We present a method of solving partial differential equations on the $n$-dimensional unit sphere using methods based on the continuous wavelet transform derived from approximate identities. We give an explicit analytical solution to the Poisson equation and to the Helmholtz equations. For the first one and for some special values of the parameter in the latter one, we derive a closed formula for the generalized Green function.

math.AP↗
arXiv2025

Green function to the biharmonic equation on $n$-dimensional sphere

Homogeneous and inhomogeneous biharmonic equation are considered on the $n$-dimensional unit sphere. The Green function is given as a series of Gegenbauer polynomials. In the paper, explicit representations of the Green function are found for some sets of parameters.

math.AP↗
arXiv2018

On the uncertainty product of spherical functions

The uncertainty product of a function is a quantity that measures the trade-off between the space and the frequency localization of the function. Its boundedness from below is the content of various uncertainty principles. In the present paper, functions over the $n$-dimensional sphere are considered. A formula is derived that expresses the uncertainty product of a continuous function in terms of its Fourier coefficients. It is applied to a directional derivative of a zonal wavelet, and the behavior of the uncertainty product of this function is discussed.

math.CA↗
arXiv2018

Uncertainty product of the spherical Gauss-Weierstrass wavelet

In this paper, asymptotic behavior of the uncertainty product of Gauss-Weierstrass wavelet is investigated. It is shown that the uncertainty product is bounded from above, a feature that distinguishes Gauss-Weierstrass wavelet among other wavelet families.

math.CA↗
arXiv2018

Uncertainty product of the spherical Abel-Poisson wavelet

In the present paper, the variances in the space and momentum domains of the spherical Abel-Poisson wavelet, as well as the limit of the uncertainty product for $ρ\to0$, where $ρ$ is the scale parameter, are computed. The values of these quantities coincide with a certain accuracy with those of the spherical Poisson wavelet with $\tfrac{1}{2}$ substituted for the order parameter.

math.CA↗
arXiv2018

Spin weighted wavelets on the sphere

In the present paper, a construction of spin weighted spherical wavelets is presented. It is based on approximate identities, the wavelets are defined for a continuous set of parameters, and the wavelet transform is invertible directly by an integral.

math.FA↗
arXiv2018

Angular multiselectivity with spherical wavelets

We construct spherical wavelets based on approximate identities that are directional, i.e. not rotation-invariant, and have an adaptive angular selectivity. The problem of how to find a proper representation of distinct kinds of details of real images, ranging from highly directional to fully isotropic ones, was quite intensively studied for the case of signals over the Euclidean space. However, the present paper is the first attempt to deal with this task in the case of spherical signals. A multiselectivity scheme, similar to that proposed for $\mathbb R^2$-functions, is presented.

math.CA↗
arXiv2018

Multiresolution on n-dimensional spheres

In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are examined. Decomposition and reconstruction algorithms for the wavelet transform are presented. Formulae for variances in space and momentum domain, as well as for the uncertainty product, of zonal functions over n-dimensional spheres are derived and applied to the scaling functions.

math.CA↗
arXiv2018

Uncertainty of Poisson wavelets

Poisson wavelets are a powerful tool in analysis of spherical signals. In order to have a deeper characterization of them, we compute their uncertainty product, a quantity introduced for the first time by Narcowich and Ward in~\cite{NW96} and used to measure the trade-off between the space and frequency localization of a function. Surprisingly, the uncertainty product of Poisson wavelets tends to the minimal value in some limiting cases. This shows that in the case of spherical functions, not only Gauss kernel has this property.

math.CA↗
arXiv2018

Uncertainty product of spherical wavelets

In the paper, asymptotic behavior of the uncertainty product for a family of zonal spherical wavelets is computed. The family contains the most popular wavelets, such as Gauss-Weierstrass, Abel-Poisson and Poisson wavelets and Mexican needlets. Boundedness of the uncertainty constant is in general not given, but it is a property of some of the wavelets from this class.

math.CA↗
arXiv2018

Poisson wavelets on $n$-dimensional spheres

In this paper, Poisson wavelets on $n$-dimensional spheres, derived from Poisson kernel, are introduced and characterized. We compute their Gegenbauer expansion with respect to the origin of the sphere, as well as with respect to the field source. Further, we give recursive formulae for their explicit representations and we show how the wavelets are localized in space. Also their Euclidean limit is calculated explicitly and its space localization is described. We show that Poisson wavelets can be treated as wavelets derived from approximate identities and we give two inversion formulae.

math.CA↗
arXiv2018

Continuous wavelet transforms on $n$-dimensional spheres

In this paper, we are concerned with $n$-dimensional spherical wavelets derived from the theory of approximate identities. For nonzonal bilinear wavelets introduced by Ebert \emph{et al.} in 2009 we prove isometry and Euclidean limit property. Further, we develop a theory of linear wavelets. In the end, we discuss the relationship to other wavelet constructions.

math.FA↗
arXiv2018

Semi-continuous and discrete wavelet frames on $n$-dimensional spheres

The paper shows that under some mild conditions $n$-dimensional spherical wavelets derived from approximate identities build semi-continuous frames. Moreover, for sufficiently dense grids Poisson wavelets on $n$-dimensional spheres constitute a discrete frame. In the proof we only use the localization properties of the reproducing kernel and its gradient.

math.CA↗
arXiv2018

Frames of directional wavelets on $n$-dimensional spheres

The major goal of the paper is to prove that discrete frames of (directional) wavelets derived from an approximate identity exist. Additionally, a kind of energy conservation property is shown to hold in the case when a wavelet family is not its own reconstruction family. Although an additional constraint on the spectrum of the wavelet family must be satisfied, it is shown that all the wavelets so far defined in the literature possess this property.

math.CA↗
arXiv2018

Directional wavelets on $n$-dimensional spheres

Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on $n$-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.

math.CA↗