Smoothness and Function Spaces Generated by Homogeneous Multipliers

Differential operators generated by homogeneous functions ψ of an arbitrary real order s > 0 ψ-derivatives and related spaces of s-smooth periodic functions of d variables are introduced and systematically studied. The obtained scale is compared with the scales of Besov and TriebelLizorkin spaces. Explicit representation formulas for ψ-derivatives are obtained in terms of the Fourier transform of their generators. Some applications to approximation theory are discussed.


Introduction
Smoothness is one of the basic concepts of analysis, having a long history.A fundamental observation is its strong connection with the decay of the Fourier coefficients of a given function.This paved the way to apply methods of Fourier analysis to the further development of smoothness concepts.Let us consider two directions of the Fourier analytic approach to describe the differentiability properties of functions.The first one is related to the scales of Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q which are constructed by means of decomposition of the Fourier series into dyadic blocks with the help of an appropriate resolution of unity see, e.g., 1-5 for periodic and nonperiodic setting .The second direction is based on the interpretation of a derivative as an operator of multiplier type.In this case one also deals with the Fourier coefficients, but not with decomposition into blocks.Following this way the concept of classical derivative was essentially extended to fractional derivatives such as Riesz and Weyl derivatives see, e.g., 6 and later on to the concept of generalized derivatives and the corresponding scale of the Stepanets classes in the one-dimensional case see 7, 8 .Both the theory of function spaces as it has been developed by the Russian S. M. Nikol'skij or German H. Triebel school and the theory of generalized smoothness elaborated by Stepanets and his coworkers have found many applications in various fields of modern mathematics.More precisely, the first direction is mainly applied to the theory of partial differential equations, computational mathematics, stochastic processes, and fractal and nonlinear analysis.The second one turned out to be important for many problems of approximation theory, in particular, constructing optimal linear approximation methods on various classes of smooth functions and obtaining approximation relations with asymptotically sharp constants.
It is well known see 5 , Ch. 3, or 4 , Ch. 2, in the nonperiodic case , for 1 < p < ∞ and s > 0, that the space F s p,2 coincides with the fractional Sobolev H s p which is related to the operator I − Δ s/2 , where Δ is the Laplace operator and I is the identity operator.
Here "relation to operator" means that H s p consists of periodic functions f in L p such that I − Δ s/2 f belongs to L p as well.In other words, we have H s p I − Δ −s/2 L p which means that H s p is characterized as the image of L p by the operator I − Δ −s/2 .With this exception both Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q are not related to any operator acting on L p in general.For this reason the scales of Besov and Triebel-Lizorkin are not well adapted to the study of problems which are directly connected with concrete operators.In contrast to these function spaces, the classes of Stepanets are related to certain operators of multiplier type which are connected to the so-called ψ, β -derivatives; see, for example 7 .In Stepanets' theory the one-dimensional case is considered only.The smoothness generator ψ is an arbitrary function satisfying some natural conditions.On the one hand, it gives quite a lot of freedom and allows the description of many interesting properties of functions which are smooth in this sense.On the other hand, such a "poor" information on ψ does not enable us to obtain explicit representation formulas for related derivatives in terms of the functions under consideration itself in place of their Fourier coefficients.This fact prevents the application of this approach to many important problems of numerical approximation.
In the present paper, we will introduce and study operators D ψ of multiplier type generated by homogeneous functions and related spaces X p ψ of periodic functions of d variables.On the one hand, homogeneity seems to be a rather general assumption.Practically all known differential operators as, for instance, the classical derivatives, Weyl and Riesz derivatives, mixed derivatives, and the Laplace operator and its fractional powers are generated by homogeneous multipliers.On the other hand, taking into account that the Fourier transform in the sense of distributions of a homogeneous function of order s is also a homogeneous function of order − d s see, e.g., 9, Theorem 7.1.6, one can derive quite substantial statements concerning the corresponding operators and related function spaces.In particular, we will prove that there is unique space X p ψ which coincides with the fractional Sobolev space H s p if ψ is homogeneous of degree s > 0 and 1 < p < ∞ Theorems 3.1 and 4.1 .However, we get infinitely many new spaces in the cases p 1 and p ∞ Theorem 3.4 .Moreover, we find an explicit representation formula for D ψ f for functions f belonging to the periodic Besov space B s p,1 if ψ is homogeneous of degree s, 0 < s < 1 Theorem 5.1 .
Let us also mention that it is aimed to extend the approach to generalized smoothness based on the introduction of operators D ψ and spaces X p ψ in further works by i finding representation formulae for operators generated by homogeneous functions of degree s ≥ 1 using higher-order differences, ii studying generalized K-functionals and their realizations, iii constructing new moduli of smoothness related to D ψ , iv studying the same concept in nonperiodic case, v investigating generalized differential equations.
The paper is organized as follows.Section 2 deals with notations and preliminaries.The basic properties of operators D ψ and spaces X p ψ are described in Section 3. Some relations between spaces X p ψ and Besov and Triebel-Lizorkin spaces are discussed in Section 4. Finally, Section 5 is devoted to the derivation of an explicit formula for D ψ in terms of the Fourier transform of the generator ψ.

Numbers and Vectors
By the symbols N, N 0 , Z, R, C, N d 0 , Z d , and R d we denote the sets of natural, nonnegative integer, integer, real and complex numbers, d-dimensional vectors with non-negative integer, integer and real components, respectively.The symbol T d is reserved for the d-dimensional torus 0, 2π d .We will also use the notations for the scalar product and the l 2 -norm of vectors and for the open and closed balls, respectively.

L p -Spaces
As usual, L p ≡ L p T d , where 1 ≤ p < ∞, T d 0, 2π d , is the space of measurable real-valued functions f f x , x x 1 , . . ., x d which are 2π-periodic with respect to each variable satisfying In the case p ∞, we consider the space C ≡ C T d p ∞ of real-valued 2π-periodic continuous functions equipped with the Chebyshev norm For L p -spaces of non-periodic functions defined on a measurable set Ω ⊆ R d we will use the notation L p Ω .

Fourier Coefficients and Fourier Transform
The Fourier coefficients of f ∈ L 1 are defined by Let E per and E per be the space of infinitely differentiable periodic functions and its dual the space of periodic distributions, respectively.The Fourier coefficients of f ∈ E per are given by where, as usual, f, g means the value of the functional f at g ∈ E per .
The Fourier transform and its inverse are given by For an element f belonging to the space of tempered distributions S S R d , which is the dual of the Schwartz space S S R d of rapidly decreasing infinitely differentiable functions, the Fourier transform is defined by setting f, g f, g , g ∈ S. 2.8

Trigonometric Polynomials
Let σ be a real nonnegative real number.By T σ we denote the space of all real-valued trigonometric polynomials of spherical order σ.It means where c is the complex conjugate to c ∈ C. Further, T stands for the space of all real-valued trigonometric polynomials of arbitrary order.Let 1 ≤ p ≤ ∞.As usual, we put for the best approximation of f in L p f ∈ C if p ∞ by trigonometric polynomials of order at most σ in the metric of L p .

Homogeneous Functions
iii ψ is homogeneous of order s; It follows that ψ ∈ S and that the restriction of Theorem 7.1.18,page 168 .

Periodic Besov and Triebel-Lizorkin Spaces
The Fourier analytical definition is based on dyadic resolutions of unity see, e.g., 5, Chapter 3 , or 4 for the nonperiodic case .Let ϕ 0 be a real-valued centrally symmetric ϕ 0 −ξ ϕ ξ for all ξ ∈ R d infinitely differentiable function satisfying

2.13
We put Clearly, these functions are also centrally symmetric and infinitely differentiable with compact support.We have

2.15
By 2.14 we obtain Combining 2.13 and 2.16 , one has In view of 2.15 and 2.17 , the system Φ {ϕ j } ∞ j 0 is called a smooth dyadic resolution of unity.
Let s > 0, 1 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞.The periodic Besov space B s p,q and the periodic Triebel-Lizorkin space F s p,q are given by cf., 5, Chapter 3 and by where the function f j , j ∈ N 0 , are defined by and ϕ j , j ∈ N 0 , are given by 2.13 and 2.14 .It is well known that Definitions 2.18 -2.21 are independent of the choice of the resolution of unity Φ.The associated norms are mutually equivalent.Therefore, we do not indicate Φ in the notation of norms and spaces.For the details, further properties, and natural extensions to parameters s ∈ R, 0 < p, and q ≤ ∞, we refer to 5, Chapter 3 .

Fourier Means
Let ϕ be a real-valued centrally symmetric continuous function with a compact support.It generates the operators F ϕ σ which are given by f is called Fourier mean of f generated by ϕ.The functions W σ ϕ in 2.23 are defined as The Fourier means describe classical methods of trigonometric approximation which are well defined for functions in L p , where 1 ≤ p ≤ ∞.They are well studied and investigated in detail in many books and papers on approximation theory; see, for example, 6, 10 .Let us also mention 5, Chapter 3 , for a treatment within the framework of periodic Besov and Triebel-Lizorkin spaces.Following 11 , we recall and state the following properties.
i The set of the norms of operators defined on L p by 2.23 is uniformly bounded, and we have Journal of Function Spaces and Applications where the positive constant c does not depend on f and σ.
If θ is given by 2.14 , then in view of 2.13 and 2.27 the functions f j which are well defined for f ∈ L 1 by 2.22 can be represented as

2.31
Moreover, using 2.16 and 2.27 we get for J ∈ N. Taking into account that ϕ 0 ∈ L 1 R d , we obtain therefrom by the convergence property 2.28 of the Fourier means the decomposition into a series of trigonometric polynomials being convergent in the space L p , 1 ≤ p ≤ ∞.

Inequalities of Multiplier Type for Trigonometric Polynomials
Let ψ ∈ H s for some s > 0. It generates the family of operators A σ ψ σ defined by on the space T of trigonometric polynomials.We have proved in 12, 13 that the inequality

ψ-Smoothness and Basic Properties
Let ψ ∈ H s for some s > 0. It generates an operator D ψ by setting which is initially defined on the space of trigonometric polynomials.The domain of definition can be extended within the spaces L p , 1 ≤ p ≤ ∞.To this end we introduce the space X p ψ which consists of functions f in L p having the property that the set is the system of the Fourier coefficients of a certain function in L p .This function will be called ψ-derivative of f in the following.By this definition we have for the Fourier coefficients of the ψ-derivative of f.Its uniqueness follows from the wellknown fact that each L 1 -function is uniquely determined by the set of its Fourier coefficients.
If f ∈ X p ψ , then the series in 3.1 with f in place of t converges in L p see the proof of Theorem 3.1 .In this sense we can reformulate We give some examples.Let d 1, s ∈ N, 1 < p < ∞, and ψ ξ iξ s .Then, D ψ is the operator of the usual derivative of order s.In this case X p ψ is the Sobolev space W s p of s − 1 -times differentiable functions f such that f s−1 is absolutely continuous, f s exists almost everywhere and belongs to L p .If d 1, s / ∈ N, and ψ ξ iξ s |ξ| s e sgn ξ• sπi /2 , then D ξ is the Weyl derivative • s of fractional order s and X p ψ is the corresponding Weyl class.For d 1 and ψ ξ |ξ|, the operator D ψ is the Riesz derivative • .Taking into account that it is the composition of the usual derivative of the first order and the operator of conjugation, we see that in this case X p ψ coincides with the space W 1 p of those functions where both the function itself and its conjugate belong to W 1 p .It is well known that this space coincides with W 1 p if and only if 1 < p < ∞.For more details concerning the derivatives and spaces mentioned above, we refer to 5-8 .
In the multivariate case d > 1 the classical Laplace operator Δ and its fractional power −Δ s/2 , s > 0, are operators of type D ψ associated with ψ ξ −|ξ| 2 and ψ ξ |ξ| s , respectively.In this case X p ψ ≡ X p −Δ s/2 coincides with the periodic version of the Bessel-potential space the fractional Sobolev space if 1 < p < ∞ considered in 14, Chapter 5 see also the next subsection .Now we study the properties of the spaces X p ψ .Recall that we have to replace Theorem 3.1.Let ψ ∈ H s , s > 0, and 1 ≤ p ≤ ∞.Then, i X p ψ is a Banach space with respect to the norm iii all spaces X p ψ with ψ ∈ H s coincide and their norms are equivalent if 1 < p < ∞.
Proof.In order to prove i we only have to check completeness because all other properties of Banach space are obviously fulfilled.Let {f n } be a Cauchy sequence in X p ψ .In view of 3.5 By completeness of L p there exist functions f, F ∈ L p , such that Using 3.3 and H ölder's inequality, we get for any k ∈ Z d and n ∈ N. In view of 3.7 the right-hand side of this inequality tends to 0 if n → ∞ .This implies Hence, f is ψ-differentiable, D ψ f F, and completeness is proved.
In order to show part ii we consider the Fourier means F ϕ 0 σ of type 2.23 -2.24 , where ϕ 0 satisfies the conditions described in Section 2 see, in particular, 2.13 .Let f ∈ X p ψ .Taking into account that for σ ≥ 0. The terms on the right-hand side tend to 0 if σ → ∞ by 2.28 .This yields the desired density of T in X p ψ .Now we prove part iii .Let 1 < p < ∞, ψ 1 , ψ 2 ∈ H s , and let f be an arbitrary function in X p ψ 2 .First we observe that in view of 2.34 and 3.1 , inequality 2.35 can be rewritten as

3.12
It is valid for 1 < p < ∞ see the comment at the end of Section 2 .Therefore, we obtain f } is a Cauchy sequence in L p by 2.28 , and there exists F ∈ L p such that By the help of 3.3 , 2.27 , and 3.14 we get For each k ∈ Z d .Hence, f belongs to X p ψ 1 and D ψ 1 f F. In order to prove that the embedding is continuous, it is enough to notice that as a consequence of 2.35 .This completes the proof.
In other words part iii of the theorem means that in the case 1 < p < ∞ there is only one space X s p of s-smooth functions.It can be characterized as and may be equipped with the norm and this embedding is continuous.
Proof.Let f ∈ X p ψ 1 .In view of 2.15 the function ϕ θψ 2 /ψ 1 , where θ is given by 2.14 , is infinitely differentiable on R d and has compact support.Hence, its Fourier transform belongs to L 1 R d and the operators F ϕ σ are uniformly bounded in L p as stated in Section 2. Using this fact as well as 3.3 , 2.31 , and the homogeneity property of ψ 1 and ψ 2 , we get

3.21
for n > m ≥ 1.Here, f j , j ∈ N, has the meaning of 2.22 .Because of s 1 > s 2 the sequence of the partial sums of the series ∞ j 1 D ψ 2 f j is fundamental in L p and there exists F ∈ L p such that

3.22
By 2.13 we have ϕ 0 k / 0 only for k 0. Thus, we obtain for any J ∈ N in analogy to 2.32 .Considering the limit process J → ∞ we find for any ν ∈ Z d with the help of 2.13 and 3.22 .Hence, f belongs to X p ψ 2 and D ψ 2 f F.
In order to prove that embedding 3.20 is continuous, it is enough to put m 1 in estimate 3.21 and to consider n to ∞.This completes the proof.
is a surjective mapping.
Proof.Let f ∈ L 0 p .We introduce functions g j , j ∈ N, by setting

3.27
for n > m ≥ 1.Hence, there exists g ∈ L p such that g L p ∞ j 1 g j .

Journal of Function Spaces and Applications
Taking into account that ϕ 0 k / 0 only for k 0 in view of 2.13 , we obtain analogously to 2.32 29 by 3.26 .As a consequence of 3.28 and 3.29 we get for ν / 0. Finally, because of f ∧ 0 0 we obtain

3.31
From 3.30 .Hence, g ∈ X p ψ , D ψ g f, and the proof is complete.
Theorem 3.3 shows that for any ψ ∈ H s an operation which is inverse to D ψ is well defined on L 0 p .The operator is called operator of ψ-integration.The Fourier series of the function

3.33
As it follows from the proof of Theorem 3.3 the series 3.33 converges in the sense of 3.26 and 3.28 .Taking in 3.27 m 1 and considering n to ∞, we obtain the boundedness of the operator I ψ in L 0 p .With the help of Theorem 3.3 we will see that in contrast to the case 1 < p < ∞, where in view of part iii of Theorem 3.1 only one space of s-smooth functions exists, in the cases p 1, ∞ there are infinitely many ways to define smoothness spaces of order s associated with homogeneous generators by means of operators of multiplier type.Theorem 3.4.Suppose that ψ 1 and ψ 2 are linear independent homogeneous functions belonging to H s , s > 0.Then,

3.34
Proof.We consider the case p 1.For p ∞ the proof is similar.Let us assume to the contrary that

3.35
Then, by Theorem 3.3 the operators where ϕ 0 has the meaning of Section 2 see also 2.13 , are well defined on the space L 0 1 .The operator L n acts in accordance with the following chain of mappings and inclusions: 3.37 By 2.23 , 2.27 , 3.3 , and 3.33 we get for n ∈ N. Here,

3.39
In view of 3.38 each operator L n is bounded in L 0 1 .Moreover, the boundedness of for any f ∈ L 0 1 .Applying now the Banach-Steinhaus principle we conclude that the operators L n are uniformly bounded in L 0 1 .This leads to the estimate where the constant c does not depend on f ∈ L 0 1 and n ∈ N. Let now t x |k|≤m c k e ikx 3.42 be an arbitrary trigonometric polynomial.We choose n > 2m.Then, it holds that ϕ 0 k/n ψ 2 k ψ 2 k for |k| ≤ m in view of 2.13 .Applying 3.41 to f D ψ 1 t, we obtain

3.43
This contradicts the statement on the nonvalidity of inequality 2.35 for p 1 pointed out at the end of Section 2. Changing the roles of ψ 1 and ψ 2 completes the proof.

ψ-Smoothness and Besov and Triebel-Lizorkin Spaces
The aim of this section is to compare the spaces X p ψ , ψ ∈ H s , with periodic Besov and Triebel-Lizorkin spaces.As we have seen already in part iii of Theorem 3.1 there is a unique space X s p if 1 < p < ∞ which has been characterized in 3.18 and 3.19 .The following theorem shows that it coincides with the classical fractional Sobolev space defined by s > 0 We put ψ * ψ 1 − η .By 2.15 we have where θ is given by 2.14 .
Let f be an arbitrary function in B s p,1 , and let f j , j ∈ N 0 , be given by 2.22 .Using 4.7 , the homogeneity property of ψ, and 2.27 and applying a Fourier multiplier theorem which can be found in 5, Theorem 3.3.4,page 150 , we obtain for n > m ≥ 1.In view of 2.18 and 4.8 we can conclude that the series ∞ j 0 D ψ f j converges in L p .Now, by standard arguments we see as in the proof of Theorem 3.1 that f belongs to X p ψ and that the first embedding in 4.5 is continuous.
In order to prove the second embedding we introduce an infinitely differentiable function ψ * ξ which coincides with ψ ξ for |ξ| ≥ 1/4 and which is not equal to 0 on R d .By 2.15 we have where θ has the meaning of 2.14 .

Journal of Function Spaces and Applications
Let f be an arbitrary function in X p ψ , and let f j , j ∈ N 0 , be given by 2.22 .Applying 4.9 , the homogeneity property of ψ, and 2.27 , we obtain for j ∈ N from 4.10 .Obviously, Hence, by 4.11 and 3.5 for some constant c > 0 and all f ∈ X p ψ .This completes the proof.
Having in mind Theorem 4.1 the embeddings 4.5 are well known in the case 1 < p < ∞.Even a better result holds.This follows also from Theorem 4.1 and the elementary embeddings 4.14 see 5, Remark 3.5.1/4,page 164 .

A Representation Formula for ψ-Derivatives
The following observations give some motivation and pave the way to find explicit representations of the operator D ψ in terms of the Fourier transform of its generator ψ ∈ H s .
For the sake of simplicity we restrict ourselves to the case 0 < s < 1.
Let ψ ∈ H s .Recall that g ∈ E per if and only if

5.6
We claim that the integral on the right-hand side of 5.6 exists for all x ∈ T d .To this end we first recall that see Section 2 ψ ∈ S is a homogeneous distribution of order − d s .Its restriction to R d \{0} can be identified with a function ψ ∈ C ∞ R d \{0} by 9, Theorem 7.1.18 .Hence, we have the estimate For brevity we use the standard notation Δ h g x g x h − g x .Now, we split The second summand is finite because of 5.7 and the boundedness of |Δ h g x | on R d .As for the first one we use the estimate and 5.7 to see that

5.10
The above arguments suggest that formula 5.6 might be true in a stronger sense.Indeed, the following theorem shows that the representation for the derivative D ψ g holds pointwise almost everyone under much weaker assumptions with respect to g. Proof.First we recall that B s p,1 ⊂ X p ψ by Theorem 4.2.Hence, the left-hand side of 5.11 makes sense.Let f ∈ B s p,1 .As is well known see, e.g., 5, Theorem 3.5.4, 12 is an equivalent norm in the Besov space B s p, 1 .Using 5.7 , 5.12 , and the generalized Minkowski inequality we obtain for the integral I x at the right-hand side of 5.11 the estimates .

5.13
Hence, the function I x belongs to L p .To prove 5.11 it is sufficient to show that the Fourier coefficients of the functions on both sides coincide.We have This is obvious if k 0 because of ψ 0 0. If k / 0, we have to use appropriate limiting arguments to circumvent the difficulty caused by the fact that ψ is not integrable in a neighbourhood of 0. We do not go into details.
We give some remarks.It is known see 15 5.17 for the fractional power of the Laplace operator.Let us mention that formulas for ψ-derivatives with s ≥ 1 can be achieved using differences of higher order.Suppose, for example, that 1 ≤ s < 2 and, in addition to the previous conditions, that ψ is real valued.Analogously to 5.6 we find

by 2 .
27 and 3.3 we get .26 where θ has the meaning of 2.14 , and we put ϕ θ/ψ.By the same arguments as in the proof of Theorem 3.2 see, in particular, 3.21 we get

2
−j k θ 2 −j k f ∧ k e ikx p any j ∈ N. The right-hand side can be estimated again by means of the Fourier multiplier theorem which can be found in 5, Theorem 3.3.4,page 150 .Using in addition the uniform boundedness of the Fourier means F θ 2 j in L p , we obtain the inequalities
s theorem.It remains to show that R d e ikh − 1 ψ h dh 2π −d ψ k , k ∈ Z d .5.15 which is valid for 0 < s < 1 and 1 ≤ p ≤ ∞.In the multivariate case we get under the same conditions with respect to s and p the representation formula−Δ s/2 f x 2π −d c d, s

D ψ g x 2π −d R d g x h − 2g x g x − h 2
g ∈ E per .Similarly to the proof of Theorem 5.1 one can show that formulas 5.19 and 5.20 are valid at least for functions belonging to the Besov spaces B s p,1 .
A complex-valued function ψ defined on R d \ {0} is called homogeneous of order s ∈ R if for t > 0 and ξ ∈ R d \ {0}.An element ψ of the space S is called homogeneous distribution of order s see, e.g., 9, Def.3.2.2,page 74 if for any t > 0 ψ, g t• t − s d ψ, g , g ∈ S. 2.12 It is well known see, e.g., 9, Theorem 7.1.16,page 167 that the Fourier transform of a homogeneous distribution ψ of order s is also a homogeneous distribution of order − s d .Let now s > 0. By the symbol H s we denote the class of functions ψ satisfying the following conditions: ψ 1 is generated by ψ 1 ∈ H s 1 and A σ ψ 2 is generated by ψ 2 ∈ H s 2 , is valid for all t ∈ T σ and σ ≥ 1 with a certain constant c independent of t and σ for all 1 ≤ p ≤ ∞ if s 1 > s 2 .If s 1 s 2 , then 2.35 is valid for 1 < p < ∞ and for arbitrary generators ψ 1 , ψ 2 .If s 1 s 2 and p 1 or p ∞, then the validity of 2.35 implies that the functions ψ 1 and ψ 2 are proportional.
p , 2.35 where A σ Both statements are known.The second identity can be found in 5, Theorem 3.5.4,page 169 .To prove the first identity and to show the equivalence of norms one can use the Fourier multipliers and the theorem of Mikhlin-H örmander see, e.g., 15, Theorem 5.2.7, page 367 , for the non-periodic version which can be transferred to the periodic case using 16, Chapter 7, Theorem 3.1 .The arguments are similar to 17, Subsections 6.2.2, 6.2.3 , or 18, Theorem 6.3.2 , where equivalent characterizations and connections between nonperiodic homogeneous and inhomogeneous Sobolev spaces are treated.We omit the details.Let η be an infinitely differentiable positive function defined on R d such that , Theorem 2.4.6, page 128 that for any s, 0 < s < 2, the restriction of the Fourier transform of ψ ξ |ξ| s to R d \ {0} can be identified with|•| s ∧ x c d, s |x| −d−s , c d, s 2 d s π d/2 Γs/2 d/2 Γ −s/2 , 5.16 combining 5.7 with 5.16 in the one-dimensional case, we obtain the well-known formula for the Riesz derivative see, e.g., 6