Slicing of radial functions: a dimension walk in the Fourier space

Abstract Computations in high-dimensional spaces can often be realized only approximately, using a certain number of projections onto lower dimensional subspaces or sampling from distributions. In this paper, we are interested in pairs of real-valued functions ( F , f ) on $$[0,\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that are related by the projection/slicing formula $$F (\Vert x \Vert ) = {\mathbb {E}}_{\xi } \big [ f \big (|\langle x,\xi \rangle | \big ) \big ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>‖</mml:mo> <mml:mi>x</mml:mi> <mml:mo>‖</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>ξ</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for $$x\in {\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where the expectation value is taken over uniformly distributed directions in $${\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . While it is known that F can be obtained from f by an Abel-like integral formula, we construct conversely f from given F using their Fourier transforms. First, we consider the relation between F and f for radial functions $$F(\Vert \cdot \Vert )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mo>‖</mml:mo> <mml:mo>·</mml:mo> <mml:mo>‖</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> that are Fourier transforms of $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> functions. Besides d - and one-dimensional Fourier transforms, it relies on a rotation operator, an averaging operator and a multiplication operator to manage the walk from d to one dimension in the Fourier space. Then, we generalize the results to tempered distributions, where we are mainly interested in radial regular tempered distributions. Based on Bochner’s theorem, this includes positive definite functions $$F(\Vert \cdot \Vert )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mo>‖</mml:mo> <mml:mo>·</mml:mo> <mml:mo>‖</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and, by the theory of fractional derivatives, also functions F whose derivative of order $$\lfloor \nicefrac {d}{2}\rfloor $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>⌋</mml:mo> </mml:mrow> </mml:math> is slowly increasing and continuous.