Approximation Theory, Computational Mathematics and Numerical Analysis in new conception of Computational (Numerical) Diameter

During the fast moving 4th industrial revolution, caused by the development of computer technology, the methods of optimal processing of information on computational tools in mathematical models acquiring special importance. The mathematical equivalent of this is proposed in 1996 [1] and immediately supported by academician of the Academy of Sciences of the USSR and Russia S.M. Nikolsky representation in the report the Russian Academy of Sciences Computational (numerical) diametry (С (N) D), the meaning of which is, we hope, a new understanding of the theory of approximations, computational mathematics and, in general, numerical analysis.
In C(N)D the initial definition is

$$\delta _{N} (\varepsilon _{N} ;D_{N} )_{Y} \equiv \delta _{N} (\varepsilon _{N} ;T;F;D_{N} )_{Y} \equiv \mathop{\inf}\limits_{\left(l^{\left(N\right)},\varphi _{N} \right)\in D_{N}} \delta _{N} \left(\varepsilon _{N} ;\left(l^{\left(N\right)},\varphi _{N} \right)\right)_{Y}, \eqno(*)$$

where
$$\delta _{N} \left(\varepsilon _{N} ;\left(l^{\left(N\right)},\varphi _{N} \right)\right)_{Y} \equiv \delta _{N} (\varepsilon _{N} ;T;F;\left(l^{\left(N\right)},\varphi _{N} \right))_{Y} \equiv $$
\small
$$\equiv\mathop{\sup}\limits_{\substack{f\in F \\ \left|\gamma _{N}^{(\tau )} \right|\le 1\left(\tau =1,....,N\right)}} \left\| Tf\left(\, \cdot \, \right)-\varphi _{N} \left(l_{N}^{(1)} \left(f\right)+\gamma _{N}^{(1)} \varepsilon _{N}^{(1)},...,l_{N}^{(N)} \left(f\right)+\gamma _{N}^{(N)} \varepsilon _{N}^{(N)} ;\cdot \right)\right\| _{Y} .$$
\normalsize
Here, the mathematical model is defined by the operator $T:F\mapsto Y$, where $X$ and $Y$ - normalized spaces of functions defined on $\Omega _{X} $ and $\Omega _{Y} $, $F\subset X$, respectively class of functions.
Numerical information $l_{N}^{\left(N\right)} \equiv l_{N}^{\left(N\right)} \left(f\right)=\left(l_{N}^{\left(1\right)} \left(f\right),...,l_{N}^{\left(N\right)} \left(f\right)\right)$ volume $N(N=1,2,3,...)$ about $f$ from the class $F$ is removed from the linear functionals defined on it $l_{N}^{(1)},...,l_{N}^{(N)} $ (in the general case, not necessarily linear). Algorithm for processing information $\varphi _{N} \left(z_{1},...,z_{N} ;\cdot \right):$ $C^{N} \times \Omega _{Y} \mapsto C$ is a correspondence, which for all fixed $\left(z_{},...,z_{N} ;\cdot \right)\, \, \in C^{N} $ as a function of $\left(\cdot \right)$ is an element of $Y$.
Writing $\varphi _{N} \in Y$ means that $\varphi _{N} $ satisfies all the conditions listed above, and $\left\{\varphi _{N} \right\}_{Y} $ will denote a set composed of all $\varphi _{N} \in Y$. Next$\left(l^{(N)},\varphi _{N} \right)$, it is a computational recovery unit for exact information for the function $f\in F$, acting according to the rule $\varphi _{N} (l_{N}^{(1)},...,l_{N}^{(N)} ;\cdot )$.
Recovery of $T(f)$ from unexact information is as follows. At first set boundaries of inaccuracy-vector $\varepsilon _{N} =\left(\varepsilon _{N}^{\left(1\right)},...,\varepsilon _{N}^{\left(N\right)} \right)$ with non-negative components. Then, the exact values $l_{N}^{(\tau )} (f)$ are replaced with a given accuracy $\varepsilon _{N}^{\left(\tau \right)} \ge 0$ per approximate value $z_{\tau} \equiv z_{\tau} (f),$ $\left|z_{\tau} -l_{N}^{\left(\tau \right)} (f)\right|\le \varepsilon _{N}^{(\tau )} (\tau =1,...,N)$, numbers $z_{\tau} \equiv z_{\tau} (f),$$(\tau =1,...,N)$ are processed by the algorithm $\varphi _{N} $ to the function $\varphi _{N} (z_{1} (f),...,z_{N} (f);\cdot )$, which will be the computing unit $(l^{(N)},\varphi _{N},\varepsilon _{N} )\equiv \varphi _{N} (z_{1} (f),...,z_{N} (f);\cdot )$, constructed from the accuracy information $\varepsilon _{N} =(\varepsilon _{N}^{(1)},...,\varepsilon _{N}^{(N)} )$.
Let $D_{N} \equiv D_{N} (F)_{Y} $ be a given set of complexes $(l_{N}^{(1)},...,l_{N}^{(N)} ;\varphi _{N} )\equiv (l^{(N)},\varphi _{N} )\equiv (l^{(N)},\varphi _{N} ;0)$, we emphasize, recovery operators "for unexact information ", as the original this circle of questions.
The entries $A\ll B$ and $A \asymp B$ respectively mean $|A|\le cB(c>0)$ and simultaneousExecution of $A\ll B$ and $B\ll A$.
The value (*) will be called the "informative power of the set of computational aggregates (complexes) $D_{N} \equiv D_{N} (F)_{Y} $ of accuracy $\varepsilon _{N} =(\varepsilon _{N}^{(1)},...,\varepsilon _{N}^{(N)} )$.
In order to shorten speech, we will say ``The computing unit \textit{$(\bar{l}^{(N)},\bar{\varphi}_{N} )\in D_{N} $} supports the lower bound $\delta _{N} (0;T;F;(\bar{l}^{(N)},\bar{\varphi}_{N} ))_{Y} \ll\vartheta _{N} $, if the inequality $\vartheta _{N} \ll\delta _{N} (0;T;F;D_{N} )_{Y} $.
Within the framework of the above notation and definitions, the problem of optimal recovery according to unexact information with the service of computing on computers,named '' Computational (numerical) diameter'', is, concludes, in the sequential solution of the following three tasks- C(N)D-1, C(N)D-2 and C(N)D-3.
Given $T,F,Y,D_{N} $ (fixed throughout the following context):
C(N)D-1. The order $ \asymp \delta _{N} (0;D_{N} )_{Y} \equiv \delta _{N} (0;T;F;D_{N} )_{Y} $ informative power set of computational aggregates $D_{N} \equiv D_{N} (F)_{Y} $ with the construction of a specific computational aggregate \textit{$(\bar{l}^{(N)},\bar{\varphi}_{N} )$} supporting order $ \asymp
$$\delta _{N} \left(0;D_{N} \right)_{Y} $.
C(N)D-2. For $\left(\overline{l}^{\left(N\right)},\bar{\varphi}_{N} \right)$ the problem of existence and finding sequences $\tilde{\varepsilon}_{N} \equiv \tilde{\varepsilon}_{N} \left(D_{N} {\rm ;}\left(\overline{l}^{(N)},\bar{\varphi}_{N} \right)\right)_{Y} \equiv \left(\tilde{\varepsilon}_{N}^{\left(1\right)},...,\tilde{\varepsilon}_{N}^{\left(N\right)} \right)$ with non-negative components- C (N) D-2-margin of error (corresponding to the optimal computational aggregates $\left(\overline{l}^{\left(N\right)},\bar{\varphi}_{N} \right)$
$$
{\kern 1pt} {\kern 1pt} \delta _{N} (0;D_{N} )_{Y} \asymp \delta _{N} (\tilde{\varepsilon}_{N} ;(\overline{l}^{(N)},\bar{\varphi}_{N} ))_{Y} \equiv
$$
$$
\equiv\mathop{\sup}\limits_ \{\| Tf(  \cdot   )-\bar{\varphi}_{N} (z_{1} (f),...,z_{N} (f);\cdot )\| _{Y} :      f\in F,    |\bar{l}_{\tau} (f)-z_{\tau} (f)|\le \varepsilon _{N}^{(\tau )} (\tau =1,...,N)\},
$$ with simultaneous execution
$$
\forall \eta _{N} \uparrow {\kern 1pt} +\infty (0<\eta _{N} <\eta _{N+1},    \eta _{N} \to {\kern 1pt} +\infty ): \overline{\mathop{\lim}\limits_{N\to +\infty}}{\delta _{N} (\eta _{N} \tilde{\varepsilon}_{N} ;(\overline{l}^{(N)},\bar{\varphi}_{N} ))_{Y} \mathord{/{\vphantom{\delta _{N} (\eta _{N} \tilde{\varepsilon}_{N} ;(\overline{l}^{(N)},\bar{\varphi}_{N} ))_{Y} \delta _{N} (0;D_{N} )_{Y}}}.\kern-\nulldelimiterspace} \delta _{N} (0;D_{N} )_{Y}} =+\infty
$$.
C(N)D-3. Set the massiveness of the marginal error $\tilde{\varepsilon}_{N} \left(D_{N} ;\left(\overline{l}^{(N)},\bar{\varphi}_{N} \right)\right)_{Y} $: is as large as possible $M_{N} \left(\overline{l}^{\left(N\right)},\overline{\varphi}_{N} \right)$ from $D_{N} $ (usually associated with source structure $\left(\overline{l}^{\left(N\right)},\bar{\varphi}_{N} \right)$) computational aggregates $\left(l^{(N)},\varphi _{N} \right)$, constructed according tofunctionals $l^{\left(N\right)} \equiv \left(l_{N}^{(1)},...,l_{N}^{(N)} \right)$ (in the general formulation is not necessarily linear), such that for each one completed
$$
\forall \eta _{N} \uparrow {\kern 1pt} +\infty (0<\eta _{N} <\eta _{N+1},    \eta _{N} \to {\kern 1pt} +\infty ): \overline{\mathop{\lim}\limits_{N\to +\infty}}{\delta _{N} (\eta _{N} \tilde{\varepsilon}_{N} ;(l^{(N)},\varphi _{N} ))_{Y} \mathord{/{\vphantom{\delta _{N} (\eta _{N} \tilde{\varepsilon}_{N} ;(l^{(N)},\varphi _{N} ))_{Y} \delta _{N} (0;D_{N} )_{Y}}}.\kern-\nulldelimiterspace} \delta _{N} (0;D_{N} )_{Y}} =+\infty .
$$
If it turns out, that in C(N)D-1 there will be more than one extreme computational aggregates, then for each of them a C(N)D-2,3 analysis is carried out, since their computational qualities are determined not only by the size of the maximum margin, but also by adaptability of the structure of the computing unit to the characteristics of the object of application.
&quot;Approximation theory&quot; and &quot;Computational Mathematics&quot; are, in fact, a replacement for the complex,in a certain sense, an object on a simple object, with constructive and computational advantages, respectively, with a mandatory assessment of the error that occurs. It seems to us that C(N)D-1 in the main should and can be a quantitative description of this verbal wording.
The theory of approximations and computational mathematics (linear aspect) in the context of C (N) D-1 is proposed to be understood as follows: with given $T,F,Y,D_{N} $ composed of all possible linear functionals over $F$and from $\left\{\varphi _{N} \right\}_{Y} $, it is required to construct a specific computational aggregate $\overline{\varphi _{N}}\left(\overline{l}_{1} \left(f\right),...,\overline{l}_{N} \left(f\right)\right)$ with a property
$$
\delta _{N} (0;T;F;D_{N} )_{Y} \asymp \mathop{\sup}\limits_{f\in F} \| Tf-\overline{\varphi}_{N} (\overline{l}_{1} (f),...,\overline{l}_{N} (f),\cdot )\| _{Y}.$$
Of course, further specification of the $D_{N} $ leads to special problems such as "‘ Approximate capabilities of a particular system "’," ‘Approximate calculation of the values of functions, integrals and other complex objects"’, etc.
A special case of approximation theory and computational mathematics is $D_{N} $ with nonlinear functionals.
And,of course,C(N)D in full can be considered as a new look at the whole Numerical analysis.