On approximation by Blackman-and Rogosinski-type operators in Banach space

In this paper we introduce the Blackmanand Rogosinski-type approximation processes in an abstract Banach space setting. Historical roots of these processes go back to W. W. Rogosinski in 1926. The new definitions given use the concept of cosine operator functions. We proved that in the presented setting the Blackmanand Rogosinski-type operators possess the order of approximation, which coincides with results known in trigonometric approximation. Applications for the Fourier–Chebyshev approximation are given as well.


INTRODUCTION
Let X be an arbitrary (real or complex) Banach space and [X] be the Banach algebra of all bounded linear operators U of X into itself.Let {P k } ∞ k=0 ⊂ [X] be a given sequence of mutually orthogonal projections, i.e.P j P k = δ jk P k (δ jk being the Kronecker symbol).Moreover, let us assume that the sequence of projections is total, i.e.P k f = 0 for all k = 0, 1, 2, ... implies f = 0, and fundamental, i.e. the linear span of ∪ ∞ k=0 P k (X) is dense in X.Then with each f ∈ X one may associate its unique Fourier series expansion with the Fourier partial sums operator As we know from trigonometric Fourier approximation, the convergence of the Fourier partial sums is not guaranteed for all f ∈ X.The improvement of that situation will be given by some matrix transformation like In this paper we introduce the Blackman-and Rogosinski-type operators in an abstract setting and find the order of approximation via a modulus of continuity (smoothness), which is defined by a general translation operator.The Blackman-and Rogosinski-type operators are interesting, because in a trigonometric case we are able to calculate precise values of their operator norms.Definition 1.A cosine operator function T h ∈ [X] (h ≥ 0) is defined by the following properties: (i) T 0 = I (identity operator), (ii) Remark 1.Let τ h ∈ [X], h ∈ R, be a translation operator, defined by the properties (i) Then T h := 1 2 (τ h + τ −h ), h ≥ 0, is a cosine operator function.
First let us consider some examples.
Classical trigonometric case.Let X = C 2π be the space of 2π-periodic continuous functions and τ h ( f , x) = f (x + h), h ∈ R, which obviously forms a semigroup translation with T = 1.Then by Remark 1 forms the cosine operator function.
It is well known that the Fourier partial sums operator π-Symmetric trigonometric case.Let X = C − 2π be the space of π-symmetric continuous functions, i.e. we suppose that f (2π − x) = f (x) and f (4π + x) = f (x) for all x ∈ R. For example, the functions y = cos kx, y = sin(k π] under a usual scalar product, we may consider the Fourier partial sums operator where ∑ ′ means that the coefficient a 0 is halved and ) tdt.
If a function f ∈ C 2π , it is obvious that τ h f ∈ C 2π as well.But it should be noted that the ordinary translation operator τ h ( f , x) = f (x + h), h ∈ R, is not suitable for the π-symmetric functions, since, for example, τ h ( sin Analogously to the periodic case we have ) ) .
Fourier-Chebyshev series.For f ∈ C [−1,1] let us consider the Fourier-Chebyshev partial sums operator where is the kth Fourier-Chebyshev coefficient and T k (u) = cos (k arccos u) is the kth Chebyshev polynomial.For this case a suitable cosine operator function (see [1,3]) is Since it can be verified that T C h (T k , x) = cos kh T k (x), we have The paper is organized as follows.Section 2 is devoted to preliminary notions like the modulus of continuity and the best approximation, but in an abstract setting using the cosine operator function.The main definitions of the Blackman-and Rogosinski-type approximation operators are introduced.
Section 3 treats the order of approximation by the Blackman-and Rogosinski-type approximation operators.
Section 4 concerns applications to the trigonometric approximation operators.
In Section 5 we consider a quite specific problem.Namely, it appears that the exact values of the operator norms of the Blackman-and Rogosinski-type trigonometric approximation operators can be calculated.
In Section 6 we apply the previous results to the Fourier-Chebyshev series.

MODULUS OF CONTINUITY, THE BEST APPROXIMATIONS, AND BLACKMAN-AND ROGOSINSKI-TYPE OPERATORS
In the present section we will define the Blackman-and Rogosinski-type approximation operators.The main idea for definitions below was inspired from the trigonometric approximation (see [2,4,5,7] and references cited there).An abstract modulus of continuity, defined by the cosine operator function, will play an important role in our paper.

Definition 2. The modulus of continuity of order k is defined via the cosine operator function by
(2.1) Remark 2. Let τ h : X → X, h ∈ R, be a translation operator and let us define another modulus of continuity by Then by Remark 1 Another quantity we need is the best approximation.
is the best approximation of f ∈ X by polynomials p n ∈ Π n .
Remark 3. Usually we may assume that for each f ∈ X there exists Let us define the Rogosinski-and Blackman-type operators.
Definition 4. The Rogosinski-type operators R n,h,a : X → X are defined via the Fourier partial sums and the cosine operator function is defined by Remark 4. The case a = 1 leads to the original Rogosinski operator R n,h : C 2π → C 2π which was introduced by W. W. Rogosinski [5] in trigonometric approximation and was afterwards elaborated by S. B. Stechkin in [6]; see also [2,7,8].
Definition 5.The Blackman-type operators B n,h,a : X → X are defined via the Fourier partial sums and the cosine operator function is defined by Remark 5.In Definition 5 the Blackman operator in the case a = 1 2 is called the Hann operator, denoted here by H n,h .If the Fourier partial sums operator is translation invariant, i.e.T h S n = S n T h , then it is easy to prove that R 2 n,h = H n,2h and B n,h,3/8 = H 2 n,h .

ORDER OF APPROXIMATION BY BLACKMAN-AND ROGOSINSKI-TYPE OPERATORS
In this section we discuss the order of approximation of the Blackman-and Rogosinski-type operators by the modulus of continuity.
Theorem 1.For every f ∈ X and all a ∈ R for the Blackman-type operators B n,h,a : X → X we have where the constant T > 0 is from Definition 1 (assumption (iii)).
Proof.Let p * n ∈ Π n be an element of the best approximation of f ∈ X.Then By definition of Θ h,a in (3.3), using the property (iii) of the translation operator in Definition 1, we obtain For the second term in the left-hand side of (3.4) we write Thus, by Definition 2 we get Substituting all expressions above in (3.2) yields the assertion of the theorem.
Remark 6.By properties of the modulus of continuity Thus, in Theorem 1 the quantities ω( f , h) and ω( f , 2h) have the same order, and one may conclude that the simplest case is when a = 1 2 .This situation will be fixed as follows.Corollary 1.Let the Hann operator H n,h : X → X be defined by the equation Then for every f ∈ X the following holds: ) The importance of the Blackman-type operators consists in the following statement.
Theorem 2. For every f ∈ X it holds that ) Proof.Similarly to the proof of Theorem 1 we proceed where Since by Definition 1 Taking a = 5 8 in (3.5), we get which together with (3.9) and (3.7) gives the assertion.
Analogous results are valid for the Rogosinski-type operators.
Theorem 3.For every f ∈ X, a ∈ R for the Rogosinski-type operators R n,h,a : X → X it holds that Again, as by Remark 6 ω( f , h) and ω( f , 3h) have the same order and g(a) = |a| + |1 − a| (a ∈ R) has its minimum value on [0, 1], we specify Corollary 2. 1.For 0 ≤ a < 1 the following relation holds:

Let us denote R n,h
The specific value a = 9 8 yields a better order of approximation.Theorem 4. Denote R n,h = R n,h,9/8 .Then we have As we may see from Theorems 1-4 and their corollaries, Definitions 3 and 4 deduce approximation processes when the right-hand sides of the given estimates tend to zero as n → ∞ and h → 0 + .This statement requires, among others, that the sequences of operator norms ∥B n,h,a ∥ [X] , ∥R n,h,a ∥ [X] have to be uniformly bounded on n and h.From this argumentation it follows that n ∈ N and h ≥ 0 should be related in some way.We will consider this problem in certain situations like in approximation by trigonometric polynomials or by the Fourier-Chebyshev series.However, in a general framework we may state as follows.The families of Blackman-and Rogosinski-type operators possess the representation Then for every a, a 0 , a 1 ∈ R we get This equality shows that for the boundedness of ∥U a ∥ [X] for arbitrary a ∈ R it is enough to know the boundedness of ∥U a 0 ∥ [X] and ∥U a 1 ∥ [X] for two specific values a 0 , a 1 ∈ R.

APPROXIMATION BY TRIGONOMETRIC BLACKMAN-AND ROGOSINSKI-TYPE OPERATORS
It appears that in classical trigonometric approximation the family of the Blackman-type operators will be uniformly bounded if h = π n+1 .By Definition 5, using (1.2), it follows that for Blackman-type operators we have where In space C 2π the ordinary translation operator τ h ( f , x) = f (x + h), h ∈ R, is well defined, moreover T = 1.The translation operator τ h : C 2π → C 2π produces the ordinary modulus of continuity given in Remark 2. Therefore, Theorem 1 yields Theorem 5.For every f ∈ C 2π and all a ∈ R for the Blackman-type operators (4.1) it holds that ) ) .
Now we make some preparations for the calculation of the norms ∥B n,a ∥ [C 2π ] .It is known [8] that the trigonometric polynomial operator (or summability operator) where the kernel function s ∈ L 1 (R) is given by we find where by (4.2) and (4.6) with sinct := sin(πt) πt .(4.9) To calculate the norm in (4.7), we will follow the scheme Let us denote the polynomial in the numerator of (4.8) by It is clear that the signs of the polynomial p considerably influence the signs of the kernel s B,a .Let us consider for this polynomial the following possibilities.
Now for (4.17) and (4.18) we find Let us prove that We have now by (4.19) Hence, by Lemma 2 we get the assertion.

EXACT VALUES OF NORMS OF TRIGONOMETRIC BLACKMAN-AND ROGOSINSKI-TYPE OPERATORS
As we saw in Section 4, in the case 0 ≤ a ≤ 3 8 the kernel function s B,a in (4.8) has sign changes only in integers.This fact allows us to calculate the norms comparatively easily. (5.1) Proof.We have to calculate the signs of s B,a in (4.8).Since p(t) ≥ 0, for signs of s B,a we find

Let us denote
(5.2) have the following operator norms: Corollary 4. The operator norms of the Rogosinski-type operators have the following numerical values:

APPROXIMATION BY ROGOSINSKI OPERATORS OF THE FOURIER-CHEBYSHEV SERIES
Our next application of the general framework in Sections 1-3 leads to the Fourier-Chebyshev approximation, introduced and elaborated in paper [3] by P. L. Butzer and R. Stens.Now the Chebyshev cosine operator function (1.4) will be used in Definition 4. Similarly to the trigonometric case, the Rogosinski-type operators have the form where Ψ a is defined by (4.4).General Theorem 3 yields Theorem 10.For every f ∈ C [−1,1] and all a ∈ R for the Rogosinski-type operators R C n,a : ) .
We have to notice here that the modulus of continuity ] ] , using a basic reference [3], we may explain the following.The operator (6.1) can be rewritten in integral form as ([3], Section 5)

CONCLUSION
We introduced the Blackman-and Rogosinski-type approximation operators using the cosine operator function.This abstract setting is useful, because now we are able to consider different approximation problems from the unique point of view.Another feature of this paper is that in the trigonometric case we computed exact values of some operator norms of defined Blackman-and Rogosinski-type approximation operators.

Lemma 2 .
The kernel function (4.8) can be represented by s B,a (t) = a (
Let f ∈ C − 2π .Then for every h ≥ 0 the cosine operator function yields T h f ∈ C −2π , where T h :=12 (τ h + τ −h ) and τ h is the ordinary translation operator.Proof.Consider f ∈ C − 2π , then for all h ≥ 0 the function For the cosine operator function (1.1) we state following Lemma 1.