Approximation of functions, linear methods

Approximation methods defined by linear operators. If in a linear normed function space a linear manifold (linear subspace) is chosen as approximating set, then any linear operator which transforms a function into a function so that

(where and are arbitrary numbers), defines a linear approximation method for functions in by means of functions in . A linear approximation method is called projective if for all in ; it is called positive if for non-negative functions .

The finite-dimensional case is the most interesting one. Here is an -dimensional subspace, and then

(1)

where is a basis of and are certain linear functionals defined on . The choice of the linearly independent system and the set of functionals is determined by the information about the function that one intends to use while constructing the linear method. If , where is a fixed system of points in the domain of definition of , and if when , while , then , , and one has an interpolation method (for instance, Lagrange's interpolation polynomial, or an interpolation spline). If is a Hilbert space of functions and are the Fourier coefficients of a function with respect to an orthonormal system , then the sum at the right-hand side of (1) yields a linear method of orthogonal projection of onto ; in this case

Thus, this method yields the best approximation of by linear combinations of the functions .

In the theory of linear approximation methods, special attention is given to problems of convergence. Let be a Banach space, let be a linearly independent system of functions on , let be the subspace generated by the first functions of this system, and let be bounded linear operators from into , . The convergence (in the sense that as ) for any function in holds if and only if: 1) the sequence of norms of the operators is bounded (see Banach–Steinhaus theorem); and 2) for all functions in a set that is everywhere dense in . These conditions are satisfied, in particular, in the space of -periodic functions , with , and the operators defined by the trigonometric Fourier sums

(2)

of . As the set on which 2) is satisfied one can take the set of all trigonometric polynomials. If is the space (of continuous functions on the whole real axis with period ) or if equals , then as (see Lebesgue constants). Consequently, there are in and in functions to which the sequence does not converge in the appropriate metric. Since in the Banach space of functions with a norm that is invariant with respect to translations the linear operator projecting onto the subspace of trigonometric polynomials of degree at most satisfies the inequality (see [3]), the divergence in and also holds for the sequence . In particular, this is true in for the sequence of Lagrange interpolation operators with any triangular matrix of interpolation nodes. Similar facts of a negative character also hold in the non-periodic case for the linear projection operators of the spaces and onto the subspaces , of algebraic polynomials of degree at most .

The divergence of the sequence (2) for some functions in made it necessary to introduce and examine various averages of Fourier sums that do not have this disadvantage. Such are, for instance, the Fejér summation method and the Bernstein–Rogosinski summation method, which are particular cases (with a specific choice of the numerical coefficients ) of polynomials of the form

(3)

Since

(4)

the averages (3) belong to the very wide class of linear approximation methods which can be represented in the form of a convolution of with some (singular) kernel whose properties (in this case, the properties of the triangular matrix of numbers ) determine the answer to the problem of convergence. If

then the sums (3) converge uniformly when is a trigonometric polynomial, and so 2) is satisfied. For the boundedness of the norm of as an operator from into , it is necessary that

and , uniformly in and . These conditions are sufficient as well for the boundedness of the sequence if the matrix is subject to some additional restrictions (for instance, convexity or concavity of the columns). Similar to (3), using the matrix of coefficients one can also construct averages by means of Fourier–Chebyshev sums (for ) and also by means of the Lagrange interpolation polynomials with nodes (in the periodic case) or with nodes on (see, for instance, [6]).

The question of convergence of positive linear operators acting from into or from into (in particular, of operators of the form (4) with a positive kernel) is solved in terms of three test functions (see [1]): For the uniform convergence of the sequence to or to it is necessary and sufficient that this occurs for the functions , and , or, respectively, for the functions , , .

The investigation of the error of approximation by a linear approximation method leads, mostly, to studies on the rate of convergence of to , to estimates of the error in terms of the difference-differential properties of the approximated function, and to answering the question of how the linear approximation method behaves when the smoothness properties (of the approximated function) are improved.

In the examination of approximating properties of the linear approximation method (1), a natural role is played by the best approximation of by elements of the subspace . The Lebesgue inequality

(5)

shows, in connection with the Jackson inequality

(where is the modulus of continuity of a function ), that, although the order of approximation by Fourier sums is slightly worse than the best approximation ( in (5) cannot be replaced by a constant), these sums are affected by any increase of the order of differentiability of the function to be approximated. For some linear approximation methods the order of approximation cannot be higher than a certain quantity, no matter how many derivatives the function may have (the phenomenon of saturation). Thus, the order of approximation by positive linear polynomial operators cannot be higher then ; for Fejér sums the saturation order is ; and for Bernstein–Rogosinski sums it is . Interpolation splines with a specific choice of the knots (nodes, joints) ensure the best order of approximation not only for the function itself, but also for some of its derivatives, depending on the degree of the polynomials out of which the spline is composed (see [7], [8]).

In some particular cases, for specific linear approximation methods exact or asymptotically-exact estimates for the errors of approximation were obtained for some classes of functions. Strictly speaking, the first non-trivial result of this kind was established by A.N. Kolmogorov, who proved in 1935 that

where , is the class of functions for which is absolutely continuous on , and for which almost everywhere . Subsequently, results of similar character were obtained for Fourier sums (and some of their averages) with respect to other important classes of functions, for instance, in terms of upper bounds on the modulus of continuity of the -th derivative (see [2]), [6] and [9]). Special interest is given to linear approximation methods (1) yielding, on some class of functions, the least upper bound of the best approximation by the subspace . Such a property for the classes , with a specific choice of the , applies to sums of the form (3). For instance, for one should take

the property also applies to interpolation splines of order and defect 1 with knots , (see [4] and also Approximation of functions, extremal problems in function classes; Best linear method).

References

[1]	P.P. Korovkin, "Linear operators and approximation theory" , Hindushtan Publ. Comp. (1960) (Translated from Russian)
[2]	V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[3]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[4]	N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[5]	V.L. Goncharov, "The theory of interpolation and approximation of functions" , Moscow (1954) (In Russian)
[6]	A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[7]	J.H. Ahlberg, E.N. Nilson, J.F. Walsh, "The theory of splines and their applications" , Acad. Press (1967)
[8]	S.B. Stechkin, Yu.N. Subbotin, "Splines in numerical mathematics" , Moscow (1976) (In Russian)
[9]	A.I. Stepanets, "Uniform approximation by trigonometric polynomials. Linear methods" , Kiev (1981) (In Russian)
[10]	N.P. Korneichuk, "Splines in approximation theory" , Moscow (1976) (In Russian)
[11]	J.R. Rice, "The approximation of functions" , 1. Linear theory , Addison-Wesley (1964)
[12]	L.L. Schumaker, "Spline functions, basic theory" , Wiley (1981)
[13]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1963)

Comments

The fact that three test functions are enough for uniform convergence of a sequence of positive linear operators from into (or from into ) can be found in, e.g., [a1], Chapt. 3, Sect. 3. The theorem is a generalization of the Bernstein theorem, and was obtained by H. Bohman [a2]. For integral operators it was obtained by P.P. Korovkin (see also [1]).

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2]	H. Bohman, "On approximation of continuous and of analytic functions" Arkiv Mat. , 2 (1952) pp. 43–56
[a3]	H. Kiesewetter, "Vorlesungen über lineare Approximation" , Deutsch. Verlag Wissenschaft. (1973)
[a4]	R.A. DeVore, "The approximation of continuous functions by positive linear operators" , Springer (1972)
[a5]	E.W. Cheney, G.R. Hobby, P.D. Morris, F. Schurer, D.E. Wubert, "On the minimal property of the Fourier projection" Trans. Amer. Math. Soc. , 143 (1969) pp. 249–258