Spherical Jump of a Function and the Bochner–Riesz Means of Conjugate Multiple Fourier Series and Fourier Integrals

We introduce the notion of spherical jump of a function of several variables at a given point with respect to a homogeneous harmonic polynomial. Here, if the function is integrable over spheres of sufficiently small radius centered at the given point and is continuous at this point, then its spherical jump at this point with respect to any homogeneous harmonic polynomial, distinct from a constant, is zero. Under certain conditions on a function of $n$ variables ($n \ge 2$) at a point where the spherical jump of this function with respect to a homogeneous harmonic polynomial $P$ is distinct from zero, we calculate the first term of the asymptotics of the spherical Bochner–Riesz means of the critical order $(n-1)/2$ of the series (integral) conjugate to the $n$-multiple Fourier series (integral) of this function with respect to the Riesz-type kernel generated by the polynomial $P$. This first term of the asymptotics contains the spherical jump of the function as a multiplicative constant.