Papers by Katerina Hadzi-Velkova Saneva
Истражувањата во рамките на овој проект ќе дадат придонес во развојот на теоријата на вејвлет и Г... more Истражувањата во рамките на овој проект ќе дадат придонес во развојот на теоријата на вејвлет и Габор анализа на асимптотските особини на дистрибуции. Истражувањата ќе резултираат со докажување на важни Абелови и Тауберови теореми, преку кои ќе се определи зависноста на асимптотското однесување на вејвлет трансформацијата, вејвлет коефициентите, short-time Фуриеовата трансформацијата (STFT) и Габор коефициентите, од локалните и не-локалните асимптотски особини на темперирани дистрибуции. Нашите оригинални резултати ќе претставуваат комплетна карактеризација на асимптотските особини на дистрибуциите преку локализацијата на нивните вејвлет и Габор коефициенти
Journal of Mathematical Analysis and Applications, Oct 1, 2020
We characterise the wave front sets via the Stockwell transform defined as a combination of the s... more We characterise the wave front sets via the Stockwell transform defined as a combination of the short-time Fourier transformation and a special class of rotations. The main results of the first part are necessary and sufficient criteria related to the directional smoothness of a tempered distribution in the cases when the space dimension n equals 1, 2, 4 or 8. In the second part, we extend the results to arbitrary n ∈ Z + .
Integral Transforms and Special Functions, Apr 1, 2009
In this article, we investigate the asymptotic behaviour at infinity of the wavelet coefficients.... more In this article, we investigate the asymptotic behaviour at infinity of the wavelet coefficients. Assuming that the distribution f ∈ S (R) has the quasiaymptotics at 0 (respectively, the S-asymptotics at infinity) related to a regularly varying function, we obtain results for the asymptotic behaviour at infinity of its wavelet coefficients. Additionally, we analyse the boundedness of the wavelet coefficients of the quasiasymptotically bounded distribution.
Abelian results for the directional short-time Fourier transform
In this paper, we study the directional short-time Fourier transform (DSTFT) of Lizorkin distribu... more In this paper, we study the directional short-time Fourier transform (DSTFT) of Lizorkin distributions. DSTFT on the space $L^{1}(\mathbb R^{n}) $ was introduced and investigated by Giv in \cite{4}. Saneva and Atanasova extended this transform on the space of tempered distributions \cite{5}. Here, we analyze the continuity of the DSTFT on the closed subspace of $ \mathcal S(\mathbb R^{n}) $, i.e. on the space $ \mathcal S_{0}(\mathbb R^{n}) $ of highly time-frequency localized functions over $ \mathbb R^{n} $. We also prove the countinuity of the directional synthesis operator on the space $ \mathcal S(\mathbb Y^{2n}) $. Using the obtained continuity results, we will define the DSTFT on space $ \mathcal S'_{0}(\mathbb R^{n}) $ of Lizorkin distributions, and prove an Abelian type result for this transform.
Bulletin of the Malaysian Mathematical Sciences Society, 2017
We improve some of the results related to the directional short-time Fourier transform by fixing ... more We improve some of the results related to the directional short-time Fourier transform by fixing the direction and extend them to the spaces K 1 (R n) and K 1 (R) ⊗U (C n) and their duals. Then, we define multidimensional short-time Fourier transform in the direction of u k for tempered distributions, directional regular sets and their complements, directional wave fronts. Different windows with mild conditions on their support show the invariance of these notions related to window functions. Smoothness of f follows from the assumptions of the directional regularity in any direction. Keywords Short-time Fourier transform • Distributions • Directional wave front Mathematics Subject Classification 46F12 • 35A18 • 42A38 Communicated by V. Ravichandran.
Направленное кратковременное преобразование Фурье и квазиасимптотика обобщенных функций
Funkcionalʹnyj analiz i ego priloženiâ, 2019
In this paper we characterize the local and non-local asymptotic properties of Schwartz distribut... more In this paper we characterize the local and non-local asymptotic properties of Schwartz distributions via the short-time Fourier transform. We provide Abelian and Tauberian type results relating the quasiasymptotic behavior of tempered distributions with the asymptotics of their short-time Fourier transforms.
Journal of Mathematical Analysis and Applications, 2010
We develop a distribution wavelet expansion theory for the space of highly time-frequency localiz... more We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S0(R) ⊂ S(R) and its dual space S 0 (R), namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in S 0 (R). A characterization of boundedness and convergence in S 0 (R) is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.
The Laplace Equation in three variable can be reduced to three ODEs by means of the Fourier metho... more The Laplace Equation in three variable can be reduced to three ODEs by means of the Fourier method. For the cases when the exact solution does not exist, or it is complicated,we apply the wavelet-Galerkin method to the ODEs. We use suitable wavelets or scaling functions that allow finding the numerical solutions of the three equations, which will form the solution of the Laplace equation.
Tauberian theorems for distributional wavelet transform
Integral Transforms and Special Functions, 2007
In this paper we investigated the asymptotic behaviour at 0 and infinity of the distributional wa... more In this paper we investigated the asymptotic behaviour at 0 and infinity of the distributional wavelet transform. Assuming that the wavelet transform 𝒲gf(b, a) has the ordinary asymptotic behaviour at 0 (resp. at infinity) with respect to both variables (resp. to the variable b), we obtained the result for the quasiasymptotic behaviour (resp. the S-asymptotics) at 0 (resp. at infinity) of the distribution f∈𝒮′(ℝ). Additionally, we proved that the distribution has the S-asymptotics at infinity equal to zero if its wavelet transform 𝒲gf(b, a) has the S-asymptotics at infinity with respect to the variable b.
Publications de l'Institut Mathematique, 2009
We analyze the boundedness of the wavelet transform of the quasiasymptotically bounded distributi... more We analyze the boundedness of the wavelet transform of the quasiasymptotically bounded distribution. Assuming that the distribution ∈ ′ (R) is quasiasymptotically or-quasiasymptotically bounded at a point or at infinity related to a continuous and positive function, we obtain results for the localization of its wavelet transform.
Functional Analysis and Its Applications, 2019
We give an Abelian type result relating the quasiasymptotic boundedness of tempered distributions... more We give an Abelian type result relating the quasiasymptotic boundedness of tempered distributions to the asymptotics of their directional short-time Fourier transform (DSTFT). We also prove several Abelian-Tauberian results characterizing the quasiasymptotic behavior of distributions in S (R n) in terms of their DSTFT with fixed direction.
Asymptotic expansion of distributional wavelet transform
Integral Transforms and Special Functions, 2006
ABSTRACT In this paper, a quasiasymptotic behaviour and a quasiasymptotic expansion at , x0&g... more ABSTRACT In this paper, a quasiasymptotic behaviour and a quasiasymptotic expansion at , x0>0 of a distribution from are defined and applied to the distributional wavelet transform.By assuming that the distribution f∈𝒮′b(ℝ) has a quasiasymptotic at b , b>0, we determined the ordinary asymptotic behaviour at 0 of its wavelet transform 𝒲gf(b, a) with respect to the variable a. We also analysed the asymptotic expansion at 0 of the wavelet transform with respect to both variables a and b (with respect to the variable a) of distributions from 𝒮′+(ℝ) (𝒮′b(ℝ)) with appropriate quasiasymptotic expansion at 0 (b ).
In this paper we provide some Abelian and Tauberian type results relating the boundary asymptotic... more In this paper we provide some Abelian and Tauberian type results relating the boundary asymptotic behavior of the short-time Fourier transform with the quasiasymptotic behavior of tempered distributions .
We give some new results related to the directional short-time Fourier transform (DSTFT) and exte... more We give some new results related to the directional short-time Fourier transform (DSTFT) and extend them on the spaces $\mathcal K_{1}(\mathbb R^{n})$ and $\mathcal K_{1}({\mathbb R})\widehat{\otimes}\mathcal U(\mathbb C^n)$ and their duals. Then, we define multi-directional STFT and, for tempered distributions, directional regular sets and their complements, directional wave fronts. Different windows with mild conditions on their support show the invariance of these notions related to window functions. Smoothness of $f$ follows from the assumptions of the directional regularity in any direction.
The Galerkin method is one of the most used methods for finding numerical solutions of ordinary a... more The Galerkin method is one of the most used methods for finding numerical solutions of ordinary and partial differential equations. Its simplicity makes it suitable for many applications. In this paper we show that the wavelet-Galerkin method is an improvement over the standard Galerkin method for ordinary differential equations.
Applicable Analysis and Discrete Mathematics, 2016
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor fram... more We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions S (R d) and distributions of exponential type K 1 (R d) onto their images.
We define and study the ridgelet transform of (Lizorkin) distributions. We establish connections ... more We define and study the ridgelet transform of (Lizorkin) distributions. We establish connections with the Radon and wavelet transforms.
Tauberian theorems for the Stockwell transform of Lizorkin distributions
Applicable Analysis
Filomat
We study the short-time Fourier transform on the space K01(Rn) of distributions of exponential ty... more We study the short-time Fourier transform on the space K01(Rn) of distributions of exponential type. We give characterizations of K01(Rn) and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.
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Papers by Katerina Hadzi-Velkova Saneva