ON ERROR FORMULAS FOR APPROXIMATION BY SUMS OF UNIVARIATE FUNCTIONS

The purpose of the paper is to develop a new method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions. It should be remarked that formulas of this type have been known only for functions defined on a rectangle with sides parallel to coordinate axes. Our method, based on a maximization process over closed bolts, allows the consideration of functions defined on a hexagon or octagon with sides parallel to coordinate axes.


Approximation by sums of univariate functions
In some applications, one may be interested in computational techniques for the error of approximation. Diliberto and Straus [6] were the first to establish a formula for E( f ,R), where R is a rectangle with sides parallel to coordinate axes. Their formula contains the supremum over all closed lightning bolts (for this terminology, see [1,8,9,11,13]). Later the same formula was established by other authors differently in rectangular case (see [13]) and for more general sets (see [9,11]). Although the formula is valid for all continuous functions, it is not easily calculable. Some authors have been seeking easily calculable formulas for the approximation error by considering not the whole space but some subsets of continuous functions. For example, Rivlin and Sibner [14] showed that for a function f (x, y) with the continuous nonnegative derivative ∂ 2 f /∂x∂y on a rectangle R = [a 1 ,b 1 ] × [a 2 ,b 2 ] the approximation error can be calculated using the formula (1.4) Babaev [2] generalized this result and proved that the formula is valid for a continuous function f (x, y) with the nonnegative difference Δ h1h2 f . Formulas of this type for some special classes of bivariate functions were established by the author in [10].
It should be said that the known techniques for obtaining the formulas of this type considerably use a geometrical structure of the rectangle R, the fact that R is a Cartesian product of two line segments.
Our purpose is to develop a new method for obtaining explicit formulas providing precise and easy computation of E( f ,Q) on some simple polygons Q with sides parallel to coordinate axes. By simple polygons we mean polygons with the total number of vertices not more than 8. This method is based on the herein developed bolts maximization process and can be used in alternative proofs of the known results from [2,10,14]. First, we show efficiency of the method in the example of a hexagon with sides parallel to coordinate axes. Then we formulate two theorems, which can be proved in a similar way, and touch some aspects of the question about the case of an arbitrary polygon with sides parallel to coordinate axes. The condition posed on sides of polygons is essential for our method. This has several reasons, which get clear through the proof of Theorem 2.1. Here we are able to explain one of these reasons: by [8,Theorem 3], a continuous function f (x, y) defined on a polygon with sides parallel to coordinate axes has an extremal element, the existence of which is required by our method. Now, let K be a rectangle (not speaking about polygons) with sides not parallel to coordinate axes. Does any function f ∈ C(K) have an extremal element? No one knows (see [8]).
In the sequel, all the considered polygons are assumed to have sides parallel to coordinate axes.

The maximization process
Let H be a closed hexagon. It is clear that H can be uniquely represented in the form where R 1 , R 2 are rectangles and there does not exist any rectangle R such that Vugar E. Ismailov 3 A bolt of lightning (see [1]) in H is a finite ordered subset p = {p 1 , p 2 ,..., p n } of H such that p i = p i+1 , each line segment p i p i+1 (unit of the bolt) is parallel to the coordinate axis x or y, and two adjacent units p i p i+1 and p i+1 p i+2 are perpendicular. A bolt of lightning p is said to be closed if p n p 1 ⊥p 1 p 2 (in this case, n is an even number). We associate each closed bolt p = {p 1 , p 2 ,..., p 2n } with the following functional: Denote by M(H) the class of continuous functions f (x, y) ∈ C(H) satisfying the condition . Then

4)
where h, r 1 , r 2 are closed bolts formed by vertices of the polygons H, R 1 , R 2 , respectively.
Proof. Without loss of generality, we may assume that the rectangles R 1 and R 2 are of the following form: Introduce the notation It is clear that (2.7) Let p = {p 1 , p 2 ,..., p 2n } be any closed bolt. We group the points p 1 , p 2 ,..., p 2n by putting First, assume that l( f , p) ≥ 0. We apply the following algorithm, hereafter called the maximization process over closed bolts, to the bolt p.
(1) p i ∈ p + and y i+1 > y i . In this case, replace the unit p i p i+1 by a new unit q i q i+1 with vertices q i = (a 1 , y i ), q i+1 = (a 1 , y i+1 ). (2) p i ∈ p + and y i+1 < y i . In this case, replace the unit p i p i+1 by a new unit q i q i+1 with vertices q i = (a 2 , (3) p i ∈ p − and y i+1 < y i . In this case, replace p i p i+1 by a new unit q i q i+1 with vertices q i = (a 1 , y i ), q i+1 = (a 1 , y i+1 ). (4) p i ∈ p − and y i+1 > y i . In this case, replace p i p i+1 by a new unit q i q i+1 with vertices (1) and (2), (3) and (4). (2.9) It is clear that after Step 1 the bolt p is replaced by the ordered set q = {q 1 , q 2 ,..., q 2n }. We say not a bolt but an ordered set because of a possibility of coincidence of some successive points q i , q i+1 (this, e.g., may happen if the first case takes place for units p i−1 p i and p i+1 p i+2 ). Exclude simultaneously successive and coincident points from q. Then we obtain some closed bolt, which we denote by q = {q 1 , q 2 ,..., q 2m }. It is not difficult to understand that all points of the bolt q are located on straight lines x = a 1 , x = a 2 , x = a 3 .
(1) q i ∈ q + and x i+1 > x i . In this case, replace the unit q i q i+1 by a new unit p i p i+1 with vertices In this case, replace q i q i+1 by a new unit p i p i+1 with vertices p i = (x i ,b 1 ), p i+1 = (x i+1 ,b 1 ). (4) q i ∈ q − and x i+1 > x i . In this case, replace q i q i+1 by a new unit p i p i+1 with ver- It is easy to see that after Step 2 the bolt q is replaced by the bolt p = {p 1 , p 2 ,..., p 2m } and l( f , q ) ≤ l( f , p ).
(2.11) Vugar E. Ismailov 5 From (2.10) and (2.11), we deduce that It is clear that each point of the set p + coincides with one of the points (a 1 ,b 1 ), (a 2 ,b 3 ), (a 3 ,b 2 ) and each point of the set p − coincides with one of the points (a 1 ,b 2 ), (a 1 ,b 3 ), (a 2 ,b 1 ), (a 2 ,b 2 ), (a 3 ,b 1 ). Denote by m i j the number of points of the bolt p coinciding with the point (a i ,b j ), i, j = 1,3, i + j = 6. By (2.6), we can write Consequently, ≤ max l f ,r 1 , l f ,r 2 , l( f ,h) .  The proof is very simple and can be obtained by integrating the inequality ∂ 2 f /∂x∂y ≥ 0 over an arbitrary rectangle [x 1 ,x 2 ] × [y 1 , y 2 ] ⊂ H and applying the above-proven theorem.

Remark 2.3.
As seen from the proof of Theorem 2.1, we considerably use the powerful apparatus-bolts of lightning and results closely connected with this notion. Direct generalization of these results (and therefore Theorem 2.1) from our case to the case of approximation of a function f (x 1 ,...,x n ) by sums n i=1 ϕ i (x i ) is quite difficult on the simple ground that there is not yet any suitable definition of a bolt of lightning in n-dimensional case. All attempts made to solve this problem were not complete and contained gaps in the proof (see [7,11,12]). It should be noted that this interesting problem is still open.

E-bolts
The main idea in the proof of Theorem 2.1-the closed bolts maximization principlecan be successfully used in obtaining formulas of type (2.4) for functions f (x, y) defined on another simple polygons. The following two theorems include the cases of some octagons and can be proved in a similar way. 3 and let Q be an octagon of the following form: where q, r 123 , r 124 , r 234 , r 24 are closed bolts formed by the vertices of polygons Q, Theorem 3.2. Let a 1 < a 2 < a 3 < a 4 , b 1 < b 2 < b 3 and let Q be an octagon of the following form: Then where r, r 12 , r 13 are closed bolts formed by the vertices of polygons R = [a 1 , Although the closed bolts maximization process can be applied to bolts of an arbitrary polygon, some combinatorial difficulties arise when grouping values at points of a maximized bolt (a bolt obtained after maximization process) as we have done it in the proof of Theorem 2.1 (see (2.13)-(2.18)). While we cannot give a complete answer to this problem, we can describe the points of a polygon F with which points of maximized bolts coincide and state a conjecture concerning the approximation error.
Let F = A 1 A 2 ··· A 2n be a polygon with sides, as mentioned in the introduction, parallel to coordinate axes. The vertices A 1 ,A 2 ,...,A 2n in the given order form a closed bolt of lightning, which we denote by r F . The length [r F ] of the bolt r F is a number of points forming r F . In our case, [r F ] = 2n.
For example, in Theorem 3.2, the octagon Q has 3e-bolts. They are r, r 12 , and r 13 . In Theorem 3.1, the octagon Q has 5e-bolts, which are q, r 123 , r 124 , r 234 , and r 24 . The A=max l( f ,h) , l f ,r 1 , l f ,r 2 , C = max l(g,h) , l g,r 1 , l g,r 2 , (4  [3] for any continuous function defined on certain domains, which are different from a polygon with the number of sides more than 4.
To prove Theorem 4.1, we need the following lemmas.
Lemma 4.4. Let X be a normed space and let F be a subspace of X. The following inequalities are valid for any element x = x 1 + x 2 from X:

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