Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

Abstract

The approximation characteristics of trigonometric sums of special type on the class of ()-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class in the uniform metric. The range of parameters in which the sums give the order of the best uniform approximation on the classes is indicated.

1. The Introduction and Problem Definition

Let be the space of continuous -periodical functions where the norm is defined by .

Let us consider the class [1] of functions such that for given and fixed sequence () of real numbers the series are the Fourier series of some function , where The function is called the -derivative of and denoted by . For , , the class coincides with Weyl-Nagy class , and for coincides with Weyl class . In the case of natural and the class is a class of periodical functions whose th derivatives nearly everywhere do not exceed unity in absolute value. If is a sequence such that then consists of infinitely differentiable functions (see [2, Chapter 1, Section 8]). The example of a sequence satisfying condition (3) is , , . In this case the class will be denoted by . If satisfies the condition consists of analytical functions regularly continuing into the strip of the complex plane.

Following [3, page 147], we set by the set of all continuous convex downwards functions , , satisfying the condition and associate each with the characteristic where is an inverse function of . By using we define the next subset of as As is shown in [3, page 153] all functions for which where , belong to . The set of such functions is denoted by . The quantity has a simple geometric interpretation. It is equal to the length of interval where the value of the function decreases by two times. Thereby it is natural to call the function the half-decay period of . The examples of from are the functions , , , , , and others. The set also includes (see [3, page 153]) the subset of all functions for which the characteristic called the modulus of half-decay, tends monotonically to infinity as . If , , , then and that is why Therefore, .

Thus, in there exist functions that tend to zero according to the power law as well as the functions that tend to zero faster than any power function. However, the example of the function , so shows that the set may not contain functions tending to zero lower than any power function.

In what follows, we assume that a sequence defining class is a restriction on the set of natural numbers of some function , , from .

We consider for an arbitrary function the following sum: where , , is a given monotonically increasing to infinity sequence of real numbers, and , , are the Fourier coefficients of the function .

When the parameters and are chosen in a certain way, the sum coincides with some classical sums such as Zygmund sum [4] (for and , ) Fejér sum [5] (for and ) de la Vallée Poussin sum [6] (for , , and ) where and Fourier sum (for ) For , the sum coincides with the so-called generalized Zygmund sum [7] (see also [8, 9]) where () is a given monotonically increasing to infinity sequence of real numbers.

The aim of the current paper is an investigation of asymptotic behavior as of the quantity where is the sum of type (11) with , , . Note that this research area for Fourier sum, de la Vallée Poussin sum, and Zygmund sum has a long history on different functional classes. It is connected with Kolmogorov, Nikol’skii, Timan, Dzyadyk, Stechkin, Korneichuk, Efimov, Telyakovskii, Stepanets, Motornii, Trigub, Rukasov, and others. For more information on this subject see, for example, [10–13].

Following Stepanets [1], we call that the solution of Kolmogorov-Nikol’skii problem is found for sum on the class if the asymptotic equality is obtained, where is some specific sequence.

In the current work, the solution of Kolmogorov-Nikol’skii problem for sum is found on and it is shown that this sum provides higher order of approximation in the uniform metric than Fourier sum, de la Vallée Poussin sum, and other classical sums do.

2. The Main Results

Theorem 1. Let , , , and . Then, as where , , and is a quantity uniformly bounded in , , and .

As is shown in [3, page 508], if and , then for the quantity of the best uniform approximation of the class by trigonometric polynomials of order not more than : the estimate is true (the notation means that there exist constants such that ). Theorem 1 and (25) lead to the following statement.

Corollary 2. Let , , , , , and . Then the order estimate holds, implying that the sum provides the order of the best uniform approximation of the class .

If , where or , then equality (22) gives the solution of Kolmogorov-Nikol’skii problem for .

From (22) and (23) we can write the following.

Corollary 3. Let the conditions of Theorem 1 be satisfied. Then, as where , , , is a quantity uniformly bounded in , , and , and the notation means that there exists a constant such that .

Taking into account that, for , , , the equality holds, from Corollary 3, we have the following.

Corollary 4. Let , , , , and . Then, as where , and is a quantity uniformly bounded in , , and .

Note that if and , then the sum provides higher order of approximation on the class than de la Vallée Poussin sum does. Indeed, it is proved in [14, page 981] (see also [15, page 130] and [16, page 10]) that where is a quantity uniformly bounded in , , , and . Comparing (30) with (32), we find that if satisfies the condition then As noted earlier, the sum for coincides with Fourier sum of order . For such a value of , from Theorem 1 we get the following.

Corollary 5. Let , , and . Then, as where , , , and is a quantity uniformly bounded in and .

Equality (35) was obtained by Stepanets (see, for example, [3, page 257]). It gives the solution of Kolmogorov-Nikol’skii problem for Fourier sum when For , (in this case and ), equality (35) takes the from Asymptotic equality (37) was established by Kolmogorov [17] (for ) and Pinkewitch [18] (for ).

Corollary 2 and equality (35) show that in the case when and is chosen such that , the sum provides higher order of approximation on in comparison with Fourier sum.

Setting in Corollary 3 and taking into account that is a generalized Zygmund sum , , we obtain the following.

Corollary 6. Let , , , and . Then, as where , , , is a quantity uniformly bounded in , , and the notation means that there exists a constant such that .

Since by definition (8) from (25), (38), and inclusion , we have This estimate follows also from Corollary 2 for .

Taking , , in (41) (in this case and , ), we get Note that estimate (42) can be obtained from more precise results of Telyakovskii [19] (see Theorem 7).

3. Proof of Theorem 1

Consider the quantity where and belongs to the set (as shown in [3, page 155], ). By Theorem 4.1 [2, page 71], the following equality holds at any point : where Let us simplify the right-hand side of (45) to obtain the principal term of the quantity . To do this, set Taking (48) and (49) into account, equality (45) can be written in the following form:

Since the class is invariant under shifts of the argument (if , then the function , , also belongs to ), for quantity (20) we have It follows from (51) that we may restrict our attention to at .

We prove the following result.

Lemma 7. Assume that , , , , and is an arbitrary sequence of real numbers satisfying the condition where is some positive constant. Then for any function , as , one has where and is a quantity uniformly bounded in , , and .

Proof. By (50) Lemma 7 will be established if we show that Since estimate (56) can be obtained in the same way as (55) we only prove estimate (55).
Assume that . Represent the quantity in the form Using for the expression obtained from (47) and (48), we have Taking into account the inclusion , the estimate and condition , it follows from (59) that As proved in [3, Chapter 5, Section 11] (see (11.9) and (11.17)), if , then Therefore, from (61) we obtain
Consider the second integral on the right-hand side of (57). To find its principal term represent the quantity in the form obtained from (58) by integrating the second integral by parts: As is easy to see, estimate (60) implies that Then by (64)
Now we consider the third integral in (57). Integration by parts in the first integral of (64) yields Since from (68) and (69) we have
Combining (63), (67), (70), and the estimate proved in [3] (see (11.31)), from (57) we get
Assume that . The following equality holds: By using representation (58) and estimate (60) it is not hard to establish that In view of (65) and (68), we have This and give us the estimate for the second integral in (73): To find the estimate of the third integral on the right-hand side of (73) we use representation (68). We have Since we hence obtain from (78) by using (71) that Combining (73), (74), (77), and (80) and taking into account (62), we get
For the proof of lemma in the case when , it is enough to apply the estimate [1, page 119] (see Lemma 5) where , to the first integral of (81). Indeed, using (82) we can write Thus, for estimate (55) follows from (81) and (83). Lemma 7 is proved.

Setting in Lemma 7 where is defined by (6) and (), and taking into account the following estimates from [3, pages 155, 156]: we obtain the following statement.

Corollary 8. Assume that , , , and . Then for any , as , the following equality holds: where and is a quantity uniformly bounded in , , and .

We now set where , , , and are selected in such a way that if , if , and if .

In the above notation the integral on the right-hand side of (87) can be represented in the form where

We claim that Making elementary transformations in (95) and taking into account that , we have Since we can write the estimate Combining (101) and (103), we obtain By (86) is bounded below; therefore, (99) follows from (104).

Since it follows from (96) that

Combining (87), (94), (99), and (100), we obtain

We proceed to the finding of the estimate of . Substituting (107) in (51), we have We shall show that where

Bearing in mind the symmetry of the function relative to of the segment , , we obtain From this we get that Since this gives us the equality Using (103), we have Comparing (114) and (115), we find that Using the notations of (97) and (98), we represent (116) in the form

Since we have Comparing (117) and (119), we obtain Thus, where is defined by (110).

In an analogous way one can prove that Relation (109) follows from (112), (121), and (122).

Comparing (108) and (109), we find

We shall show that the equality sign can be put in (123). Denote by the -periodic function such that where is defined by (90). Obviously, such a function exists. According to Subsection 7.2 of [2, pages 136, 137] in , , there exists a function such that is its -derivative. From (107) and (109), we have Since , it follows from (123) and (125) that if , , , , and , then where , , and is a quantity uniformly bounded in , , and .

Theorem 1 is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported in part by the Ukrainian Foundation for Basic Research (Project no. GP/36/068).