Approximation via Hausdorff operators

Abstract Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $
 .


Introduction
e classical Hausdorff operator is defined, by means of a kernel φ, as and, as is shown first in [ ] (see also [ ] or [ ]), such an operator is bounded in L (R) whenever φ ∈ L (R).
In the last two decades, various problems related to Hausdorff operators have attracted a lot of attention. e number of publications is growing considerably; to add some of the most notable, we mention [ , , , , , ]. ere are two survey papers: [ ] and [ ]. In the latter, as well as in [ ], numerous open problems are given.
e Hausdorff operator ( . ) is expected to have better Fourier analytic properties than f. For example, in general, the inversion formula f (x) = π Rf (y)e i x y d y does not hold for f ∈ L (R); in order to "repair" this, one can consider some transformation of the function f or of its Fourier transform. In relation to the Hausdorff operator, we will consider integrals of the form R (H φf )(y)e i x y d y.

Approximation via Hausdorff operators
Here we analyze not this Hausdorff operator but a more general one, apparently first considered in [ ] (see also [ ]). Given an odd function a such that a(t) is decreasing, positive, and bijective on ( , ∞) (so that both a and a possess inverse functions in such an interval), we define It is clear that ( . ) corresponds to ( . ) with a(t) = t − , and one can easily derive the corresponding results from the general ones. Moreover, we consider some such particular cases as examples.
ere is one more reason for considering general Hausdorff operators: they provide a proper basis for future multidimensional extensions (see, for instance, [ ] and [ ], where those operators were introduced independently). Such multidimensional operators have been extensively studied in Lebesgue and Hardy spaces. We refer the reader to [ , , ] for further details. e consideration of these "alternative" transformations such as ( . ) requires the development of a parallel theory to Fourier integrals. In this paper, we address three basic issues of approximation theory applied to (generalized) Hausdorff operators.
(i) To find the operator T such that the integrals of the type approximate T f as N → ∞ (in the L p norm), for reasonable choices of φ (here some assumptions on f and φ are needed in order for (H φ,af )(y) to be well defined; see the discussion at the beginning of Section ). As we will see, the operator T is by no means the identity operator, but the dual operator of H, denoted by H * , and formally defined by the relation (ii) To study the rate of convergence to H * f of the partial integrals (iii) To modify ( . ) in a way that allows us to to derive a method for approximating f in the L p norm (rather than approximating H * f , as in (i) and (ii)).
In particular, the problem of exploiting Hausdorff operators in approximation is raised. Indeed, application of analytic results in approximation seems to be the most convincing proof of their usefulness. is work is the first attempt to understand what kind of approximation problems may appear in the theory of Hausdorff operators and to solve some of them. e results obtained will open new lines in both the theory of Hausdorff operators itself and approximation theory. e difference between Hausdorff means and more typical multiplier (convolution) means, which comes from the difference between dilation invariance for the former and shi invariance for the latter, leads not only to new results but also to novelties in the methods.
A. Debernardi and E. Liflyand e structure of the paper is as follows. In the next section, we being with certain preliminaries, we formulate the main results. In Section , we prove the main results. Section is devoted to presenting some examples of operators and their approximation estimates. A er several works on the boundedness of the Hausdorff operators on various function spaces, this paper is the first application of Hausdorff operators to the problems of constructive approximation. In particular, we compare the obtained results with their traditional counterparts (approximate identities given by convolution type operators). Finally, in Section , we give concluding remarks, and in particular, we show that some regularity of the kernel φ is needed in order to obtain good approximation estimates.
We denote by is the usual modulus of continuity. We will also write A ≲ B to denote A ≤ C ⋅ B for some constant C that does not depend on essential quantities. e symbol A ≍ B means that A ≲ B and B ≲ A simultaneously.

Main Results
First of all, let us discuss some boundedness properties of the Hausdorff operator in Lebesgue spaces, in order for H * ( f ) (and also the Hausdorff operator in ( . )) to be well defined. We will always assume that f ∈ L (R), so thatf is well defined, andf ∈ L ∞ (R). On the other hand, a sufficient condition for the operator H * to be bounded on (moreover, if φ ≥ almost everywhere, then such a condition is also necessary; see the recent paper [ ] and also [ ]). Similarly, a sufficient condition (and necessary whenever φ ≥ a.e.) for the Hausdorff operator to be bounded on L p (R) is that then H φ,af is well defined as a function from L max{ , p ′ } (R). us, we will always assume that f ∈ L (R) ∩ L p (R) and that ( . ) holds. For further results on boundedness (and also Pitt-type inequalities) of Hausdorff operators, we refer the reader to [ ].

Approximation via Hausdorff operators
We give some more observations before stating our main results. It is easy to check by substitution that we have Let us now define the partial integrals By substitution, it is easy to see that ese observations make clear that (H Nf )ˇis a good candidate to approximate H * f (informally, letting N → ∞ in ( . ) we obtain ( . )). We will prove that this is actually the case, at least in the L p setting.
Our main results concerning approximation of adjoint Hausdorff operators read as follows.

eorem .
For where we take the convention p = if p = ∞, and furthermore, A. Debernardi and E. Liflyand e fact that the adjoint Hausdorff operator of a function is approximated may be unsatisfactory in principle, as one would rather approximate the function itself. However, approximating a function instead of its adjoint Hausdorff operator is also possible as a consequence of the following observation. For φ ∈ L (R) and a(t) as in the introduction, one has is gives a natural way of approximating f through Hausdorff operators by using More precisely, we have the following theorem.

Remark .
In order for the right-hand sides of ( . ) and ( . ) to be finite, one should assume that φ vanishes at a fast enough rate as t → ∞, or even more, that it has compact support. e latter is the case for the Cesàro operator (where φ = χ ( , ) ), which we discuss in more detail in Section , along with other examples.

Proofs
First of all, we give pointwise estimates for which will be the starting points for all subsequent estimates.

Lemma .
For any x ∈ R, Proof To prove ( . ), we apply rather straightforward estimates. Indeed, as desired. In the last inequality we use that a possesses an inverse on ( , ∞) (and therefore also on (−∞, ), since it is an odd function), and moreover, ( a ) − (t) = a( t) − on ( , ∞). ∎ Note that by ( . ), Also, by ( . ), we can write, for any

Lemma .
For any x ∈ R, Proof By ( . ) and ( . ), we have the equality sin a(t)s s ds dt.
A. Debernardi and E. Liflyand e proof now follows the same lines as that of Lemma . , with the only difference being that in the above integral the term, We now proceed to the proofs of the main theorems.

Proof of eorem .
We treat the cases ≤ p < ∞ and p = ∞ separately. For the case p = ∞, it suffices to estimate the two terms on the right-hand side of ( . ) in the L ∞ norm. For the first one, we have As for the second term on the right-hand side of ( . ), we have Collecting all the estimates, we get where the right-hand side is uniform in x.
Let us now prove the case ≤ p < ∞. Using ( . ), we get

Approximation via Hausdorff operators
Note that if p = , the factor on the le -hand side can be taken to be (in fact, such a factor appears due to the inequality (a + b) p ≤ p (a p + b p ), for a, b ≥ and p > ). On one hand, applying Minkowski's inequality twice, we get Since On the other hand, applying Minkowski's inequality again, we obtain

A. Debernardi and E. Liflyand
Collecting all the estimates, we derive where the factor on the le -hand side is omitted in the case where p = . e proof is complete. ∎

Proof of eorem .
First of all, note that the case p = ∞ follows trivially from eorem . and the fact that ω( f ; δ) = ω(τ y f ; δ) for every y ∈ R. We now show the case ≤ p < ∞. By Lemma . , If p = , the factor on the le -hand side can be omitted, similarly as in the proof of eorem . . Now, applying Minkowski's inequality twice, we estimate where the last inequality follows from the fact that ω( f ; δ) p is increasing in δ. On the other hand, applying Minkowski's inequality again, we obtain Putting all the estimates together, we finally obtain ( . ). ∎

Examples
We now obtain approximations of functions by means of certain specific Hausdorff operators. We shall give bounds for the approximation error explicitly in L p , ≤ p ≤ ∞, in each case, which will follow from eorem . .
In the first place, we consider a general Hausdorff operator under some assumptions on the kernel φ (besides the assumptions from eorem . ). We suppose without loss of generality that a(t) > for t ∈ ( , ∞), that φ is compactly supported, say on [−T, T], and φ ∈ L ∞ (R) (note that the Cesàro operator, given by a(t) = t and φ = χ ( , ) , satisfies these conditions). en, on one hand, On the other hand, Now, the substitution s → a(t) yields so we conclude that for any ≤ p ≤ ∞, A. Debernardi and E. Liflyand by eorem . (recall that ω( f ; δ) ∞ = ω( f ; δ)). If, furthermore, a(t) = t, then for ≤ p ≤ ∞, (recall also that in the case p = , the estimate on the right-hand side can be multiplied by the factor ). To the best of our knowledge, no approach through Hausdorff operators has been considered in approximation problems so far, and therefore even the basic estimate ( . ) is new in this respect.

Approximation via the Cesàro Operator
e Cesàro operator C given by a(t) = t and φ(t) = χ ( , ) (t) [ , ] is the prototype Hausdorff operator H φ,a . In this case, its adjoint operator is also referred to as the Hardy operator. We have It readily follows from ( . ) that and in the case p = ∞, we obtain a Dini-type estimate Note also that for p = , condition ( . ) does not hold, so we have to restrict ourselves to the case < p ≤ ∞. In particular, we can conclude the following corollary.

Approximation via Hausdorff operators
(ii) If f is continuous and ∫ ω( f ;t) t dt < ∞, then F N converges uniformly to f on R as N → ∞. In particular,

Remark .
For < q ≤ ∞, ≤ p ≤ ∞, and ≤ s < , the Besov seminorm (defined via the modulus of continuity) is We refer the reader to [ , § . . , eorem ] for the description of Besov seminorms in terms of moduli of continuity. Note that in Corollary . , the assumption that is equivalent to saying that the Besov seminorm B p, of f is finite.
We shall now compare the approximation estimates from Corollary . with those for approximate identities.

Comparison: Cesàro Operators and Approximate Identities
Since the Cesàro operator is the prototype example of Hausdorff operator, it is instructive to compare the obtained approximations with the classical ones given by approximate identities for convolutions. A family of functions {C r } r> defined on R is called an approximate identity if ( ) sup r C r L (R) < ∞, and ( ) for every δ > , e following is well known [ , eorem . . ].
As an example of an approximate identity satisfying ( . ), we have the family of functions where C(x) is the Fejér kernel on the real line, From now on, we assume that the approximate identities we consider satisfy condition ( . ).
Comparing eorem A and Corollary . , we readily see that the latter requires further assumptions in order to guarantee L p convergence (p < ∞), namely that the seminorm f B p, (R) is finite (cf. Remark . ). However, when restricted to certain classes of functions, the approximation rates become the same, or even better.
As classes of functions, we consider Lip p α = Lip p α(R) with < α ≤ , and ≤ p ≤ ∞, which consists of the functions f satisfying Note that Lip α = Lip ∞ α is the class of usual Lipschitzα continuous functions on R, i.e., those satisfying For f ∈ Lip p α, < α < , and ≤ p ≤ ∞, it is known that any approximate identity {C r } yields the approximation rate , while for α = , an additional logarithm appears: In the case of the Cesàro operator, Corollary . yields, for any < p < ∞ and f ∈ Lip p α,

Approximation via Hausdorff operators
with all the estimates valid for the range < α ≤ . Note that these approximation rates are the same as those for approximate identities when restricted to functions f ∈ Lip p α with < α < (compare with ( . )), and are actually better than their counterparts in the case α = (compare with ( . )), in the sense that the extra logarithm from ( . ) does not appear. us, in the case α = , the "Hausdorff "approximation improves the classical convolution approximations in the sense of rate of convergence.

Approximation via the Riemann-Liouville Integral
For α > , the Riemann-Liouville integral is defined as A rescaled version of this operator can be easily obtained as an adjoint Hausdorff operator. Indeed, for a(t) = t and φ α (t) Note that if we formally consider α = in the definition of I α , we recover the Cesàro operator.
Using eorem . , we approximate f (x) by cf. ( . ). Note that by the observation made in ( . ), we will obtain the same convergence rates via the Riemann-Liouville integral as those we obtain via the Cesàro operator. So, for continuous f, we have while for f ∈ L p (R) with < p < ∞ (note that for p = condition ( . ) does not hold, so we have to exclude such a case), we have by Corollary . and ( . ).

Final Remarks
We conclude with a couple of remarks: first, we show that one can use the same approach to approximate the Hausdorff operator (instead of its adjoint) applied to a function. Secondly, we show that we cannot expect any good approximations of Hausdorff operators if the kernel φ does not decay fast enough at infinity.

Approximation of Non-adjoint Hausdorff Operators
One can also approximate the Hausdorff operator instead of its adjoint, if one considers the adjoint Hausdorff averages in the approximant. More precisely, it is also possible to approximate H f (x) by which, by substitution, is easily seen to equal Since for any t ≠ , one has A similar estimate to that of Lemma . can now be proved.

Lemma .
For any x ∈ R,

Approximation via Hausdorff operators
Proof e proof is essentially the same as that of Lemma . , as desired. ∎ By means of the pointwise estimate from Lemma . , it is possible to obtain approximation results analogous to eorem . , where the Hausdorff operator, rather than its adjoint, is approximated. e details are essentially the same and are thus omitted.

A Hausdorff Operator with Slowly Decaying φ: the Bellman Operator
Let us see what happens if we try to approximate an adjoint Hausdorff operator with slowly decaying φ. We consider the particular example of the Bellman operator B (which is nothing more than the adjoint Cesàro operator C * ). Its adjoint B * is defined by letting a(t) = t and φ(t) = t − χ ( ,∞) (t) in ( . ): It is clear that we cannot use the methods from Section in order to approximate functions, since the hypothesis φ ∈ L (R) is not satisfied in this example. What is more, not even the basic assumption ( . ) from eorem . is satisfied for any ≤ p ≤ ∞. Nevertheless, we now try to use the approximation estimates from eorem . (heuristically, since the hypotheses of eorem . are not met) just to illustrate their bad behaviour for functions φ that do not decay fast enough. As the approximant for B * , we take x − s t ds dt.

A. Debernardi and E. Liflyand
For ≤ p < ∞, the estimate from eorem . yields while in the case p = ∞, i.e., in this case we cannot guarantee any convergence on the L p norm by using our estimates, even for well-behaved functions f. As was pointed out in Remark . , this is because in order to obtain useful estimates from eorem . , one should assume that φ is of compact support, or that it decays fast enough as t → ∞. For the adjoint Cesàro operator, the functions φ has some decay, but it is not fast enough. Also note that the estimate ( . ) is not good, as the right-hand side is infinite for nonconstant functions.