Application of spherical functions to a problem of the theory of quadratic forms

The problem regarding the number of integral points on multidimensional ellipsoids is investigated with the aid of modular forms. In the paper we consider the simplest special case of the following problem: one considers a multidimensional sphere and as a domain on it one selects a "cap.’’ The precise result is formulated in the following manner: let $r_\ell(n)$ be the number of the representations of $n$ by a sum of $\ell$ squares, $0<A<1$; then for even $\ell\geq 6$ we have
$$ \sum_{-A\leq\frac{x}{\sqrt{n}}\leq A}r_{\ell-1}(n-x^2)=r_\ell(n)\left(K_\ell(A)+O\left(n^{-\frac{\ell-2}{2(\ell+1)}+\varepsilon}\right)\right); $$
for $\ell=4$ we have
$$ \sum_{-A\leq\frac{x}{\sqrt{n}}\leq A}r_3(n-x^2)=r_4(n)\left(K_4(A)+O\left(n_1^{-\frac{1}{5}+\varepsilon}\right)\right), $$
where $n=2^\alpha n_1$, $2^\alpha\,\|\,n$; the expression for $K_\ell(A)$, $\ell\geq4$, is given in the paper. It is also shown that one can refine somewhat the results on the distribution of integral points on multidimensional ellipsoids, obtained by A.V. Malyshev by the circular method, remaining within the framework of the same methods.