The paper is devoted to the study of a self-adjoint extension \({\mathcal L}\), on the entire axis \(\mathbb{R}= (-\infty, +\infty)\), of the Schrödinger operator \(Hu= -u''+ q(x)u\) defined on \(\mathbb{R}\) with the potential \(q(x)\) to be uniformly locally integrable. For an arbitrary \(p \in [1,2]\) the expansion of any function \(f(x)\in L_p(\mathbb{R})\) into a Fourier integral is proved to equiconverge with the \({\mathcal L}\)-corresponding spectral expansion \(\sigma_\lambda (x,f)\) of this function. Moreover, for the spectral function \(\theta (\lambda,x,y)\) of the extension \({\mathcal L}\) the author obtains the following two estimates: \[ \left|\theta (\lambda,x,y) -{\sin \sqrt \lambda(x-y) \over\pi (x-y)} \right|=O(1) \tag{1} \] which is jointly uniform with respect to \((x,y)\) for \(x\in\mathbb{R}\) and \(y\in\mathbb{R}\); \[ \left|\theta(\lambda,x,y) -{\sin \sqrt\lambda (x-y) \over \pi(x-y)} \right|_{Lq (\mathbb{R})} =O(1) \tag{2} \] which is valid for any \(q\in [2,+\infty]\) and uniform with respect to \(x\in\mathbb{R}\), where the norm is specified at the point \(y\).