Geodesic
The Abel summation of the inverse Fourier transform of the homogeneous functions in $R^n$
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2019), pp. 98-107

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As is well known, the most commonly used functions on a line are powerful functions. A multidimensional analogue of power functions is homogeneous functions, which look like $\theta (\tau )|t|^\alpha$ and have an arbitrary function on a unit sphere additionally to the parameter $\alpha$. The inverse Fourier transform for these functions results in restrictions for an order of $\alpha$. One approach to improve convergence is Abel summation. Abel summation formulas for inverse Fourier transform of homogeneous functions have been derived in the article, which look like $\theta (\tau )|t|^\alpha$, $\tau \in S^{n-1}=\{t \in \mathbb{R}^n: |t|=1\}$ for various function spaces on a unit sphere.
Keywords: Abel summation formula, homogeneous functions.
Mots-clés : inverse Fourier transform 
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     author = {S. V. Arhipov},
     title = {The {Abel} summation of the inverse {Fourier} transform of the homogeneous functions in $R^n$},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {98--107},
     publisher = {mathdoc},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2019_4_a6/}
}
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S. V. Arhipov. The Abel summation of the inverse Fourier transform of the homogeneous functions in $R^n$. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2019), pp. 98-107. http://geodesic.mathdoc.fr/item/VTPMK_2019_4_a6/