Least p-Variances Theory

As a result of a rather long-time research started in 2016, this theory whose structure is based on a fixed variable and an algebraic inequality, improves and somehow generalizes the well-known least squares theory. In fact, the fixed variable has a fundamental role in constituting the least p-variances theory. In this sense, some new concepts such as p-covariances with respect to a fixed variable, p-correlation coefficient with respect to a fixed variable and p-uncorrelatedness with respect to a fixed variable are first defined in order to establish least p-variance approximations. Then, we obtain a specific system called p-covariances linear system and apply the p-uncorrelatedness condition on its elements to find a general representation for p-uncorrelated variables. Afterwards, we apply the concept of p-uncorrelatedness for continuous functions particularly for polynomial sequences and find some new sequences such as a generic two-parameter hypergeometric polynomial of 4F3 type that satisfy such a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an approximation for p-variances, an improvement for the approximate solutions of over-determined systems and an improvement for the Bessel inequality and Parseval identity. Finally, we generalize the notion of least p-variance approximations based on several fixed orthogonal variables.Comment: 85 pages, 1 figure and 1 tabl

A finite class of orthogonal functions generated by Routh-Romanovski polynomials

It is known that some orthogonal systems are mapped onto other orthogonal systems by the Fourier transform. In this article we introduce a finite class of orthogonal functions, which is the Fourier transform of Routh-Romanovski orthogonal polynomials, and obtain its orthogonality relation using Parseval identity

On weighted Adams-Bashforth rules

One class of the linear multistep methods for solving the Cauchy problems of the form $y\u27=F(x,y)$, $y(x_{0})=y_{0}$, contains Adams-Bashforth rules of the form $y_{n+1}=y_{n}+hsum_{i=0}^{k-1} B_i^{(k)} F(x_{n-i},y_{n-i})$, where ${ B_i^{(k)}} _{i = 0}^{k - 1}$ are fixed numbers. In this paper, we propose an idea for weighted type of Adams-Bashforth rules for solving the Cauchy problem for singular differential equations,[A(x)y\u27+B(x)y=G(x,y), quad y(x_0)=y_0,]where $A$ and $B$ are two polynomials determining the well-known classical weight functions in the theory of orthogonal polynomials. Some numerical examples are also included