About 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation $n!$. He gave a good formula which asymptotic to Since then hundreds papers have given alternative proofs his result and improved upon it, including notably by Burside, Gosper, Mortici. However, Srinivasa Ramanujan remarkably better formula. Hirschhorn Villarino nice proof R...

In this paper we characterize the approximation schemes that satisfy Shapiro’s theorem and we use this result for several classical approximation processes. In particular, we study approximation of operators by finite rank operators and n-term approximation for several dictionaries and norms. Moreover, we compare our main theorem with a classical result by Yu. Brundyi and we show two examples o...

nonstandard finite difference schemes for the black-scholes partial differential equation preserving the positivity property are proposed. computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the black-scholes equation. unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

In this letter, the numerical scheme of nonlinear Volterra-Fredholm integro-differential equations is proposed in a reproducing kernel Hilbert space (RKHS). The method is constructed based on the reproducing kernel properties in which the initial condition of the problem is satised. The nonlinear terms are replaced by its Taylor series. In this technique, the nonlinear Volterra-Fredholm integro...

We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) n-term approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f ; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q...

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ 1, 2 based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation ...

We study n-term wavelet-type approximations in Besov and Triebel–Lizorkin spaces. In particular, we characterize spaces of functions which have prescribed degree of n-term approximation in terms of interpolation spaces. These results are applied to identify interpolation spaces between Triebel–Lizorkin and Besov spaces.