‎Given a frame of a separable Hilbert space $H$‎, ‎we present some‎ ‎iterative methods for solving an operator equation $Lu=f$‎, ‎where $L$ is a bounded‎, ‎invertible and symmetric‎ ‎operator on $H$‎. ‎We present some algorithms‎ ‎based on the knowledge of frame bounds‎, ‎Chebyshev method and conjugate gradient method‎, ‎in order to give some‎ ‎approximated solutions to the problem‎. ‎Then we i...

‎In this paper‎, ‎two iterative methods are constructed to solve the operator equation $ Lu=f $ where $L:Hrightarrow H $ is a bounded‎, ‎invertible and self-adjoint linear operator on a separable Hilbert space $ H $‎. ‎ By using the concept of frames of subspaces‎, ‎which is a generalization of frame theory‎, ‎we design some  algorithms based on Galerkin and Richardson methods‎, ‎and then we in...

In a paper published posthumously, Pavel Samuilovich Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space. In this paper we prove that the Urysohn univeral space is hyperconvex.

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0, 1] of functions from [0, 1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0, 1]. It is shown that the closed linear span of an isometri...

The polynomial v(x) allows us to determine the points in E(k̄) that lie in the kernel of α. Indeed, we have kerα = {(x0, y0) ∈ E(k̄) : v(x0) = 0} ∪ {0}. If E1 is defined by y 2 = f(x) = x3 + Ax + B, then we get one point in kerα for each root of v that is also a root of f (these are points (x0, 0) of order 2), two points for every other distinct root of v (since α(x0, y0) = 0 implies α(x0,−y0) = ...

We demonstrate that a reproducing kernel Hilbert space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

Let X be a real separable Hilbert space. Q linear, bounded, positive and compact operator on let A : Dom ( ) ⊆ → self-adjoint generating strongly continuous semigroup . F (smooth enough) function { W t } ≥ 0 -valued cylindrical Wiener process. For any α , we are interested in the mild solution x of semilinear stochastic partial differential equation d = + > ; ∈ its associated transition (0.1) P...

We study the problem of approximating functions of d variables in the average case setting for a separable Banach space Fd equipped with a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure takes the form of a Gaussian that depends on shape parameters γl. We stress that d can be arbitrarily large. Our approximation error is defined in the L2 norm, and we study the minima...