Abstract
We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle for instance the heat-diffusion semigroup as a translation. With the given examples, Kolmogorov–Riesz characterization of compact sets in certain \(L^p_\mu \) spaces is given. Pego-type characterizations are also derived. Finally, for some examples, the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.
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1 Introduction
In 1938, Delsarte introduced the notion of generalized translation; see [13]. His starting point was the modification of Taylor’s formula by the eigenfunctions of an ordinary differential operator: \(D_xf(\lambda , \cdot )=\lambda f(\lambda ,x)\). The Taylor series of \(f=f(\lambda , z)\) around \(z_0\) at y, with the notation \(x=y-z_0\), can be expressed as
where \(D_x\varphi _0(x)=0\) and \(D_x\varphi _k(x)=\varphi _{k-1}(x)\) (\(\varphi _{-1}=0\)). Moreover, \(\varphi _k(0)=0\) if \(k>0\), and \(\varphi _0(0)=1\).
He illustrated his idea with the following two examples:
\(D_x(f)=f'\), \(f(\lambda ,x)=e^{\lambda x}\), \(\varphi _k(x)=\frac{x^k}{k!}\) and
\(D_x(f)=f''+\frac{2\alpha +1}{x}f'\), \(f(\lambda ,x)=j_\alpha (i\sqrt{\lambda } x)\), \(\varphi _k(x)=\frac{\Gamma (\alpha +1)}{\Gamma (k+1)\Gamma (k+\alpha +1)}\left( \frac{x}{2}\right) ^{2k}\); and introduced the translation operator below:
The convergence of the series above implies that \(T^t_xf(\lambda ,\cdot )=f(\lambda ,x)f(\lambda ,t)\). Examination of the second example leads to the next chain of ideas. Denoting by \(u(x,t):=T^t_xf\) the construction ensures that the translated functions are solutions to the next Cauchy problem
The initial study of the properties of generalized translation is due to Delsarte [13] and Levitan; see, e.g., [34]. Braaksma and Snoo in a series of papers dealt with the problem of introducing general translations via certain hyperbolic Cauchy problems; see, e.g., [10, 11]. Examination of product formulae with respect to classical orthogonal polynomials led to definition of Laguerre and Jacobi translations; see, e.g., [18, 20, 26].
General translation has widespread applications. Estimation of the operator norm leads to a maximum principle with respect to the hyperbolic problem in question. It gives certain convolution structures; see, e.g., [6, 32]. It also leads to estimation of p-Christoffel function; see, e.g., [4, 5]. As in the standard case approximation theoretic problems, as smoothness, best approximation or approximation by Cesaro means can be examined by general translation; see, e.g., [27, 37]. By general translation, compactness criteria can be derived too; see [31].
Here, we extend the notion of translation as solutions to Cauchy or Dirichlet problems in parabolic and elliptic cases as well. This approach allows to handle, e.g., the heat-diffusion semigroup as a translation. Although several different examples are listed and investigated below, this one is highlighted, because unlike the old and the other new examples, this one inherits the comfortable semigroup property of the standard translation, i.e., \(T^{t_1}T^{t_2}=T^{t_1+t_2}\). Moreover parabolicity allows to draw conclusions in d-dimension, while the variable of the translation is one-dimensional. We show that the general translations introduced below possess the properties of the standard one.
The paper is organized as follows. In the next section, different types of examples of translation are given. In Sect. 3, the notion of “regularity with respect to compactness” is introduced and is applied to derive Kolmogorov–Riesz-type characterizations in certain \(L^p\)-spaces. The rest of this section is devoted to show that all the listed translations are regular. In Sect. 4, Pego-type characterizations of compact sets of \(L^2\) are derived by convolution method. Section 5 deals with the approximation theoretic aspect of translation that is with modulus of smoothness and K-functional.
2 The translation operator
We define the translation operator as follows.
Definition 2.1
Let \(\Omega =I\times J\subset {\mathbb {R}}^d\times {\mathbb {R}}_{+}\). Let L be a linear partial differential operator of order at most two and take a function \(f(x)\in C(I)\) or \(f\in L^p_\mu (I)\), where \(\mu \) is a Radon measure on I. Define translation as the next linear operator: the translation of f
where u(x, t) is the solution to
where the last equality is meant in \(\sup \)-norm or in p-norm.
Remark 2.2
All of our examples can be given by appropriate integral transformations as well. The corresponding kernel functions are denoted by \(W_t(x,y)\), \(K_t(x,y)\), etc. according to the standard notation. Here, \(t>0\), and \(x,y\in {\mathbb {R}}^d\), \(d\ge 1\). That is
It also makes the solution well defined.
Below different types of examples are introduced. Of course, by the same chain of ideas, several further examples can be constructed. Here are the ones we study subsequently.
2.1 Heat-diffusion semigroup associated with Hermite functions
This semigroup, from different point of views, is investigated by several authors; see, e.g., [1, 28, 41] and the references therein.
In \({\mathbb {R}}^d\), the eigenfunctions of the d-dimensional harmonic oscillator (Hermite operator)
are the d-dimensional Hermite functions
where \(\nu =(n_1, \dots ,n_d)\) \(n_i\in {\mathbb {N}}\) and
(\(H_k\) are the Hermite polynomials; cf. [42].) The associated heat-diffusion semigroup is given by its kernel function defined on \({\mathbb {R}}^d\times {\mathbb {R}}^d \times {\mathbb {R}}_+\)
Thus, for and appropriate f, denoting by \(\tilde{f}(y):=f(y)e^{-\frac{y^2}{2}}\)
Example 2.3
Denoting by \(u(x,t):=T^tf(x)\), if \(\tilde{f}\in L^p({\mathbb {R}}^d)\), \(1\le p \le \infty \), then
and
cf. [41, Proposition 2.5 and Theorem 2.6].
2.2 Further Poisson integrals: elliptic and parabolic equations
Following the previous track of thoughts translation of a function in \(f\in L^p({\mathbb {R}})\) can be defined by Poisson integrals. First, we take the simplest elliptic and parabolic equations on the upper half-plane and define translations by standard convolution.
In the next example, \(T_x^tf\) is a harmonic function on the (open) upper half-plane with limit f on the real line and with limit zero at infinity, cf. e.g., [30], that is the solution to this Dirichlet problem
It is given by the Poisson integral below.
Example 2.4
Let \(f\in L^p({\mathbb {R}})\), \(1\le p <\infty \), or \(f\in C_b({\mathbb {R}})\)
where \(C_b({\mathbb {R}})\) stands for continuous and bounded functions on the real line.
Considering the Cauchy problem
the next translation can be defined by the corresponding Poisson integral.
Example 2.5
Let \(f\in L^p({\mathbb {R}})\) (\(1\le p \le \infty \))
We continue with an elliptic example again, where the Poisson integral is not of convolution type.
Let \(P_n^{(\lambda )}(\cos \vartheta )\) be the nth ultraspherical polynomial (\(\lambda >-\frac{1}{2}\), \(\vartheta \in [0,\pi ]\)) orthogonal with respect to \(\textrm{d}\mu (\vartheta )=\sin ^{2\lambda }\vartheta \textrm{d}\vartheta \); see, e.g., [42]. An \(f \in L^p_\mu \) with some \(1 \le p \le \infty ([0,\pi ])\) has an expansion \(f\sim \sum _{k=0}^\infty a_k P_k^{(\lambda )}(\cos \vartheta )\) with \(a_k=\gamma _k\int _0^\pi f(\vartheta )P_k^{(\lambda )}(\cos \vartheta )\textrm{d}\mu (\vartheta )\). In [35], the authors examined the next Poisson integral of f
where \(0<r<1\), \(x=r\cos \vartheta \), \(t=r\sin \vartheta \). The Poisson kernel is
Then, u(x, t) satisfies the differential equation
Since \(\Vert f(r,\vartheta )-f(\vartheta )\Vert _{\mu ,p}\rightarrow 0\) as \(r \rightarrow 1\) if \(1\le p<\infty \) and also for \(p=\infty \) if f is continuous, in view of Definition 2.1, we have the next example. As above, the closed form of the Poisson kernel implies the definition below.
Example 2.6
Let \(D=[0,\pi ]\times [0,1)\), \(f \in L^p_\mu ([0,\pi ])\)
where
Considering the next differential operator
with eigenfunctions
where
(\(L_n^{(\alpha )}\) is the standard Laguerre polynomial; see, e.g., [42]), we define a translation by the solution to the parabolic Cauchy problem
\(x,t\in (0,\infty )\), \(\alpha \ge -\frac{1}{2}\), cf. [12].
Example 2.7
where
where \(I_\alpha (z)=\sum _{k=0}^\infty \frac{\left( \frac{z}{2}\right) ^{2k+\alpha }}{\Gamma (k+1)\Gamma (k+\alpha +1)}\) is the modified Bessel function.
2.3 Fourier method: hyperbolic equation
The first—well-known—examples are given by hyperbolic equations generated by Sturm–Liouville-type operators; see, e.g., [13, 34]. Consider the operator
If
denoting by \(u(x,t)=\varphi (x)\varphi (t)\), u(x, t) fulfils the equation below
In Laguerre and Bessel cases, \(q(x)=\frac{2\alpha +1}{x}\), where \(\alpha \ge -\frac{1}{2}\) and \(r(x)=x^2\) or \(r(x)=0\), respectively. The eigenfunctions of \(D_x\) are the Laguerre functions \({\mathcal {L}}_n^{(\alpha )}(x)=\frac{n!\Gamma (\alpha +1)}{\Gamma (n+\alpha +1)}L_n^{(\alpha )}(x^2)e^{-\frac{x^2}{2}}\), and the Bessel functions \(j_\alpha (\lambda x)\), where \(L_n^{(\alpha )}(x)\) and \(j_\alpha (\lambda x)\) stand for the classical Laguerre polynomials and the entire Bessel functions, respectively. These examples are studied by several authors; see, e.g., [4,5,6, 26, 37].
In Jacobi case with \(\alpha \ge \beta \ge -\frac{1}{2}\), \(\alpha > -\frac{1}{2}\) by the same argument, \(q(x)=\frac{(\beta +1)\cos x-\beta }{\sin x}\) and \(r(x)=\frac{\alpha (\alpha -2\beta +1)\cos x}{4(1-\cos x)}\). The eigenfunctions are the Jacobi functions, \({\mathcal {P}}^{(\alpha ,\beta )}_n(x)=c_nP^{(\alpha ,\beta )}_n(\cos x)\sin ^\alpha \frac{x}{2}\), where \(P_n^{(\alpha , \beta )}(x)\)-s are the classical Jacobi polynomials. For Jacobi translation, see, e.g., [3, 20].
In the listed cases, the eigenfunctions of the Sturm–Liouville operators \(\{u_k\}_{k=0}^\infty \) form Schauder bases in the corresponding (weighted) spaces, that is, if the initial condition is given by \(f(x)\sim \sum _{k=0}^\infty a_ku_k(x)\), then the solution to the Cauchy problem can be given as \(T^tf(x)=u(x,t)\sim \sum _{k=0}^\infty a_ku_k(x)u_k(t).\)
Thus, translation generated by a hyperbolic equation is as follows:
if
Here, \(L_h\) is given by (2.11), where \(I={\mathbb {R}}_+\) in Laguerre and Bessel cases and in Jacobi case \(I=(0,\pi )\).
It is pointed out in a series of papers that Bessel, Laguerre, and Jacobi translations are bounded operators; see the cited papers above and the references therein. The method described above implies the symmetry of the translation derived by a Sturm–Liouville equation; see [10, 11].
Next, we give an example by Jacobi functions. Let \(\alpha \ge \beta \ge -\frac{1}{2}\), \(\alpha > -\frac{1}{2}\) again and let us denote by \(\varrho =\alpha +\beta +1\), and now, let \(\tau \in {\mathbb {R}}\). Let
With \(q(x)=\frac{w'}{w}(x)\), \(\varphi _\tau ^{(\alpha ,\beta )}(x)\) fulfils the differential equation
where
are the Jacobi functions. Thus, the translation is defined as it follows; cf., e.g., [21, 38, 39] and the references therein.
Example 2.8
Let f be a suitable even function on \({\mathbb {R}}\). \(T^tf(x)=u(x,t)\) if
where
3 Regularity with respect to compactness
In this section, we derive Kolmogorov–Riesz-type compactness criteria in certain \(L^p\)-spaces. For the original theorem, see, e.g., [2]. This theorem has several extensions to different function spaces with standard translation; see, e.g., [7, 15, 24, 25, 29, 40] and with Bessel and Laguerre translations, see [31]. In all but one of our examples, the translation is not symmetric. This implies that one of the criteria of compactness has to be splitted into two different assumptions; see Definition 3.4 below.
We start with some notation. Let \(\mu \) and \(\nu \) be Radon measures on I and J, respectively, and \(1\le p<\infty \). \(\Vert f\Vert _{\mu (x),p}=\left( \int _I|f|^p\textrm{d}\mu \right) ^{\frac{1}{p}}\). \(T^0f(x)=f(x)\).
We define the “norm” of the translation operator \(M_T\) as
We introduce the next notation: Let \(a>0\), \(M_0>0\) fixed, \(B_a:=\{\Vert x\Vert \le a\}\cap I\), or \(B_a:=(-a,a)\cap J\); \(A:=\int _{B_a}1\textrm{d}\nu (t)\)
Definition 3.1
A translation, \(T^t_x\), is regular with respect to p, \(\mu \), \(\nu \) if it fulfils the properties below.
-
(T1)
There is a dense set \({\mathcal {E}}_1\subset L^p_\mu (I)\), such that for all \(g\in {\mathcal {E}}_1\), \(0<t<M_0\), and \(\varepsilon >0\), there is a \(\delta =\delta (\varepsilon , M_0, g)>0\), such that for all \(0\le |h| \le \delta \)
$$\begin{aligned} \left( \int _I \left| \left( T^{t+h} g(x)-T^{t} g(x)\right) \right| ^p\textrm{d}\mu (x)\right) ^{\frac{1}{p}} <\varepsilon . \end{aligned}$$(3.2) -
(T2)
There is a dense set \({\mathcal {E}}_2\subset L^p_\mu (I)\), such that for all \(g\in {\mathcal {E}}_2\), and positive numbers \(\varepsilon \), a, R, there is a \(\delta =\delta (g,\varepsilon ,a,R)>0\), such that if \(0\le |h| \le \delta \), then
$$\begin{aligned} \left| M_{a,R}g(x+h)-M_{a,R}g(x)\right| <\varepsilon , \hspace{4pt}\hspace{4pt}x, x+h \in B_R. \end{aligned}$$(3.3) -
(T3)
There is a finite constant, c(a, R), so that for all \(f \subset L^p_\mu (I)\)
$$\begin{aligned} |M_{a,R}f(x)| \le c(a,R)\Vert f\Vert _{\mu ,p}, \hspace{4pt}\hspace{4pt}\forall \hspace{4pt}x\in I. \end{aligned}$$(3.4)
Remark 3.2
If \(T_x^y=T_y^x\) and \(\mu =\nu \), then (3.2) implies (3.3), and if \(M_T<\infty \), it also implies (3.4). Indeed
which ensures (3.3). (3.4) can be obtained in the same way.
Definition 3.3
A set \(K\subset L^p_\mu (I)\) is equivanishing, or we say it fulfils property \(\mathbf{P_a}\) if, for all \(\varepsilon >0\), there is an \(R>0\) such that for all \(f\in K\)
Definition 3.4
A set \(K\subset L^p_\mu (I)\) is equicontinuous in mean if it fulfils the next properties.
- \(\mathbf{P_{b_1}}\):
-
: For all \(\varepsilon \) and \(M_0\) positive numbers, there is a \(\delta >0\) (independent of f), such that for all \(t\in B_{M_0}\), \(0\le |h| \le \delta \) and \(f\in K\) (3.2) is satisfied.
- \(\mathbf{P_{b_2}}\):
-
: For all \(\varepsilon >0\), there is a \(\delta =\delta (a,R)>0\) (independent of f), such that for all \(0\le |h| \le \delta \) and \(f\in K\), (3.3) is satisfied.
Theorem 3.5
With the notation above, let \(T=T^t_x\) be a bounded translation in (3.1)-sense and suppose that it is regular with respect to p, \(\mu \), \(\nu \). Let \(K\subset L^p_\mu (I)\) be a bounded set. Then, K is precompact if and only if it is equivanishing and equicontinuous in mean.
Proof
First, we prove that if K is bounded, equivanishing, and equicontinuous in mean and the translation has the property T3, then K is precompact.
For an arbitrary \(\varepsilon >0\), let R be chosen according to \(\mathbf{P_a}\). Let \(a>0\) be fixed and will be chosen later, and define
Then, T3 and the boundedness of K imply that \(F_{a,R}\) is (uniformly) bounded. According to \(\mathbf{P_{b_2}}\), \(F_{a,R}\) is equicontinuous; thus, it is precompact.
Let \(M_af_1, \dots , M_af_n\) be an \(\varepsilon _1\)-net in \(F_{a,R}\), where \(\varepsilon _1=\frac{\varepsilon }{(\mu (B_R))^{\frac{1}{p}}}\). We show that \(f_1, \dots ,f_n\) is a \(5\varepsilon \)-net in K. Recalling the definition of R and A
Considering \(\mathbf{P_{b_1}}\), if a is small enough, \(\Vert M_{a,R}f-f\Vert _{\mu ,p}\le 2\varepsilon \) for all \(f\in K\). Let \(f\in K\) be arbitrary. Selecting \(f_i\), such that \(|M_{a,R}f(x)-M_{a,R}f_i(x)|\le \varepsilon _1\)
Thus, by triangle inequality
On the other hand, suppose that K is precompact. Let \(\varepsilon >0\) be arbitrary and \(\varphi _1, \dots ,\varphi _n\) be an \(\frac{\varepsilon }{2}\)-net in K. As \(C_0\), that is the set of compactly supported continuous functions, is dense in \(L^p_\mu (I)\), there are \(g_1, \dots , g_n \in C_0\), so that for all \(f \in K\), there is an i such that \(\Vert g_i-f\Vert _{\mu ,p}<\varepsilon \). Then, R is appropriate with respect to \(\mathbf{P_a}\) if \(B_R\) contains the supports of \(g_i\), \(i=1,\dots n\).
Since T is bounded, selecting \(g_1, \dots , g_n \in {\mathcal {E}}_1\) as above, triangle inequality together with T1 ensures \(\mathbf{P_{b_1}}\).
Similarly, selecting \(g_1, \dots , g_n \in {\mathcal {E}}_2\) as above, T2, T3 and the triangle inequality imply \(\mathbf{P_{b_2}}\).\(\square \)
Remark 3.6
For compactness, it is enough to assume that the initial condition, (2.2) is fulfilled uniformly. The importance of the extension above will be shown in the next section.
In the rest of this section, we show that all the translations listed above are regular in sense of Definition 3.1. At first, we make some observations which will be useful in parabolic cases below and in the last section.
Remark 3.7
For the next observation, let us write the differential equations of the listed examples in the next form
\(T^tf(x)=u(x,t)\) can be defined by an integral transformation, that is
Thus, the kernel function fulfils the differential equation above. As the kernel function is symmetric at least in x and \(\xi \), that is
we have
These observations imply the next lemma.
Lemma 3.8
Let g be a smooth function on I and assumes that its derivatives disappears quickly enough at the boundary of I. Then, in the listed cases
Proof
Recall, that in our one-dimensional examples
Integrating by parts and considering the boundary condition, we have
If \(w\equiv 1\), it is more direct. Let us see, for instance, the heat-diffusion semigroup. Recalling that \(I={\mathbb {R}}^d\)
Corollary 3.9
Let \(D_{(1)}=\frac{\partial }{\partial t}\) and let g be as above. Then
Proof
Taking into consideration (3.1), we get the estimation.
Lemma 3.10
The translation given in Example 2.3 is regular and \(M_T=1\).
Proof
The boundedness of the operator by a constant C is proved in [41, Theorem 2.6]. It can be shown similarly that \(M_T=1\). Indeed, according to Mehler’s formula and [28, Remark 2.5]
Thus
and since \(\Vert K_t\Vert _1=1\)
\(1\le p \le \infty \).
Let \({\mathcal {E}}_1= C^2_0\subset L^p({\mathbb {R}}^d)\). In view of (3.6), (T1) is fulfilled.
We prove property (T2) by similar arguments. Considering the uniform convergence of (2.5) for a \(t>0\), the recurrence and derivation formulae for Hermite functions imply that
Thus
Let \(f\in C^1_0({\mathbb {R}}^d)\), \({\textrm{supp}}f\subset B_r\), \(x\in B_R\)
To prove (T3), let us recalling (3.7). Then, we have
\(\square \)
Lemma 3.11
The translation given in Example 2.8 is regular and \(M_T=1\).
Proof
For boundedness, see [18, Lemma 5.2].
As it is mentioned above, in [11], it is pointed out that a translation generated by a hyperbolic equation is necessarily symmetric. It can be expressed by an integral transform whose kernel is symmetric in its three variables; see [18]. Thus, according to Remark 3.2, it is enough to prove property (T1). Since f is even, we can take \(I={\mathbb {R}}_+\). Let \({\mathcal {D}}=C^1_0({\mathbb {R}}_+)\subset L^p_\mu ({\mathbb {R}}_+)\) the dense set in question, where
cf. (2.13). Let \(f\in C^1_0({\mathbb {R}}_+)\) and let \(f(z)=g(\cosh 2z)\). In view of [18, (5.1)] and [18, (4.16)]
where
and
It can be readily seen that
Since \(g\in C^1_0({\mathbb {R}}_+)\) too, suppose that \({\textrm{supp}}g(z)\subset B_R\). Recalling that \(0<t<M_0\) \({\textrm{supp}}g_{t,r,\psi }(x)\subset B_{ C(M_0,R)}\). Considering that \(\frac{\partial }{\partial y}g(x,t,r,\psi )\) is bounded and \(m(r,\psi )\) is a bounded measure, one can conclude that (T1) fulfils with \(\delta = c \varepsilon \).\(\square \)
Lemma 3.12
The translations given by Examples 2.4 and 2.5 are regular and \(M_T=1\) in both cases.
Proof
The proof is based on the standard convolution structure.
First, let us observe that the kernel functions are positive and \(\int _{\mathbb {R}}K_{e,p}(x,t)\textrm{d}x=1\) for all \(t>0\), which gives the operator norm.
In both cases, let \({\mathcal {E}}_1={\mathcal {E}}_2=C_0({\mathbb {R}})\). In harmonic case by the standard substitution \(\frac{\xi -x}{t+h}=\tan \varphi \), where \(h>0\) and \(h=0\) in the different integrals, we have
which is small if \(\alpha \) is close enough to \(\frac{\pi }{2}\).
Since g is uniformly continuous for fixed \(\alpha \) and \(\varepsilon \) we can choose h, \(|h|<1\), such that \(|h\tan \varphi |\) is small enough to be the integrand in II small, less than \(\varepsilon \), say.
Then, recalling that g is compactly supported and \(|t| \le M_0\), there is an R, such that the support of \(g(x+(t+h)\tan \varphi )-g(x+t\tan \varphi )\) is in \([-R,R]\). , \(II\le \varepsilon (2R)^{\frac{1}{p}}.\)
Similarly, in parabolic case, let \(u=\frac{\xi -x}{2b\sqrt{t+h}}\). Then
Recalling that g is compactly supported, the p-norm inside is bounded; thus, if R is large enough, I is small. The uniform continuity of g implies that II is small if h is small enough.
Turning to the proof of (3.3) and (3.4), we define \(\nu (t)\). If \(1<p<\infty \) and in parabolic case also for \(p=1\), let \(\textrm{d}\nu (t)=\textrm{d}t\), and if \(p=1\), in the elliptic case, let \(\textrm{d}\nu (t)=\sqrt{t}\textrm{d}t\). Then, the same replacements imply the results. Indeed, let us see the elliptic case first
In the first and third integrals, the boundedness of the integrand, in the second one, the uniform continuity of g ensures the required estimates.
To prove (3.4), let \(1<p<\infty \)
If \(p=1\)
In the parabolic case
Again, in the first integral, we refer to boundedness; in the second one to the uniform continuity of g.
Let \(1 \le p<\infty \). Then
which is (3.4).\(\square \)
Lemma 3.13
The translation defined in Example 2.6 is regular and \(M_T=1\).
Proof
For boundedness, see [35, Theorem 2].
Let our dense sets \({\mathcal {E}}={\mathcal {E}}_i\), \(i=1,2\) be the polynomials, that is
Let \(P\in {\mathcal {E}}\).
which is (T1). (T2) can be shown similarly. Let \(\nu (r)=\lambda (r)\), the Lebesgue measure on [0, 1) and \(P\in {\mathcal {D}}\)
In view of (2.9) \(P(r,\vartheta ,\varphi )>0\), and if \(r\le \frac{1}{8}\), say, then it is also bounded. Thus, if \(a\le \frac{1}{8}\), \(1\le p<\infty \)
where \(\Vert \cdot \Vert _{\mu ,\infty }:=\Vert \cdot \Vert _{\infty }\).\(\square \)
Lemma 3.14
The translation defined in Example 2.7 is regular and bounded.
Proof
For boundedness, see [12, Theorem 2.2].
The dense sets are the set of polynomials. By the notation (2.10), it is as follows:
Let \(f\in {\mathcal {E}}\), \(f(x)=\sum _{k=0}^n c_k \varphi _k^{(\alpha )}(x)\).
Then, \(T^tf(x)=\sum _{k=0}^n e^{-(2k+\alpha +1)t}c_k \varphi _k^{(\alpha )}(x)\). Thus
that is h can be chosen appropriately depending on f.
Similarly, since \(x\in B_R\)
is small with an appropriate h depending on f.
To prove the third property, we consider the next estimation of the kernel function; see [16, (2.4)]
Let \(g\in L^p(I)\)
The consequence of Theorem 3.5 is the next one.
Corollary 3.15
Let \(K\subset L^p_\mu (I)\) be bounded. K is precompact if and only if it is equivanishing and equicontinuous in mean with T defined in Examples 2.3, 2.4, 2.5, 2.6, 2.7 and 2.8.
4 Regularity with respect to an integral transform I: Pego-type theorems
In [36], Pego characterized the compact sets of \(L^2({\mathbb {R}}^d)\) by Fourier transform. This result was extended to different spaces and to Abelian groups; see, e.g., [22, 23] and the references therein. A Pego-type theorem by Laplace transform was proved in [33]. These results based on standard translation. The extension of Pego’s theorem to Bessel and Laguerre translations and transforms is given in [31]. The main ingredients of this type of theorems are a translation, a generated convolution, and a corresponding integral transform.
4.1 Convolution
The study of convolution structures with general translations dates back to the 70s and it is a widely studied topic; see, e.g., [6, 9, 18,19,20] for Jacobi, [26] and [32]. These general convolutions similarly to the standard ones are defined as follows. For f and g, appropriate functions
All the convolutions in the listed papers are based on symmetric translations which are related to hyperbolic equations. This symmetry ensures algebraic structures with respect to the convolution in question, since the symmetry of the kernel of the translation in its three variables implies the commutativity and associativity of the convolution; see the references above. In parabolic and elliptic cases, the kernels of the translations are symmetric only in two variables; thus, the associativity fails.
To prove Pego-type theorems, the next relations of convolution, integral transform, and translation are necessary
and
where \({\mathcal {I}}\) is the integral transform and \(\psi \) is an appropriate function. The translation is regular with respect to the integral transform if it possesses the properties above.
4.2 Pego-type theorems by standard convolution
Pego-type characterization of compactness can be given in cases of Examples 2.4 and 2.5.
In elliptic case, we characterize compactness of sets in \(L^2({\mathbb {R}}_+)\) and consider cosine transformation; in parabolic case, the sets are in \(L^2({\mathbb {R}})\) and the corresponding transformation is the standard Fourier one.
The transformation pairs are normalized as follows:
where \({\mathcal {I}}=C\) or \({\mathcal {I}}={\mathcal {F}}\), the cosine or the Fourier transform with \(k(xz)=\cos (xz)\) or \(k(xz)=e^{-ixz}\), respectively. First, the actions between translations and transformations are derived. Suppose that f is in the Schwartz class, and in elliptic case, we assume that it is even
cf. [8, 1.2 (13)]. By similar calculation, supposing that \(z\ge 0\)
In the parabolic case, we have
On the other hand
As it is well known
Let \(K\subset L^p(I)\) (\(1\le p \le 2\)). Denote \(\hat{K}\subset L^{'}p(0,\infty )\), \(\hat{K}={\mathcal {I}}(K)\). With this notation, we have the next Pego-type theorem.
Theorem 4.1
Let \(1\le p \le 2\), \(K\subset L^p(I)\) a bounded set.
Let us consider Examples 2.4 and 2.5.
Assume that K satisfies \(\mathbf{P_a}\). Then, \(\hat{K}\) fulfils \(\mathbf{P_{b_1}}\) and \(\mathbf{P_{b_2}}\).
On the other hand, if K satisfies \(\mathbf{P_{b_1}}\), then \(\hat{K}\) fulfils \(\mathbf{P_a}\).
Proof
Let \(g(\cdot ,t)=e^{-|\cdot |t}\) or \(g(t,\cdot )=e^{-b^2t(\cdot )^2}\), and f as above
In view of \(\mathbf{P_a}\), the second term is small. Since \(x \in B_R\), \(g(x,\cdot )\) is uniformly continuous. Thus, considering that K is bounded, \(\mathbf{P_{b_1}}\) is satisfied for dense set uniformly and so for \(\hat{K}\)
where \(h(u)=\frac{1}{A}\int _{B_a}g(u,t)\textrm{d}\nu (t)\) is uniformly bounded in u, and if \(u\in {\mathbb {R}}\setminus B_R\)
in both cases. Thus
which is small by \(\mathbf{P_a}\), considering that \(p'>1\), and
Since the interval is bounded, h can be chosen, such that the third factor be small and so, as above, \(\hat{K}\) fulfils \(\mathbf{P_{b_2}}\).
To prove the opposite direction, we use the convolution below in cosine transform case, and the standard one in Fourier transform case. For cosine transform, supposing that f, g are even
Assume that K fulfils \(\mathbf{P_{b_1}}\). Let \(k_t(x)=K_{t,e}(x)\) or \(k_t(x)=K_{t,p}(x)\), \(x\in {\mathbb {R}}\), \(y> 0\). Choose R so large that \(|{\mathcal {I}}(k_t)(\eta )|\le \frac{1}{2}\) if \(\eta \in {\mathbb {R}}{\setminus } B_R\)
Thus, \(\mathbf{P_{b_1}}\) implies \(\mathbf{P_a}\).\(\square \)
The next corollary implies that equicontinuity in Examples 2.4 and 2.5 sense is equivalent with the standard one from this point of view.
Corollary 4.2
Let \(K\subset L^2(I)\) be a bounded set and \(\hat{K}\) the Fourier or the cosine transform of K, respectively. Then, K is precompact if and only if K and \(\hat{K}\) are equicontinuous in mean in sense of Examples 2.5 and 2.4, respectively.
4.3 A Pego-type theorem by general convolution
In Laguerre and Bessel cases, Pego-type theorems are proved in [31]. Below, we prove a similar theorem with Jacobi transform. As the translation in the original and dual spaces is different, this situation is more complex than the Bessel case.
Jacobi transform is defined as it follows; see [17, Proposition 3]. Recalling the measure \(\textrm{d}\mu (x)=\frac{1}{\sqrt{2\pi }}w(x)\textrm{d}x\) (see (2.13) and (3.8)), let \(f\in L^2_\mu ({\mathbb {R}}_+)\) and \(\lambda \in {\mathbb {R}}_+\)
where the integral is convergent in \(L^2_\nu ({\mathbb {R}}_+)\) with \(\textrm{d}\nu (\lambda )=\frac{1}{\sqrt{2\pi }}v(\lambda )\textrm{d}\lambda \), where
\({J}: f\mapsto \hat{f}\) is a linear and norm-preserving map of \(L^2_\mu ({\mathbb {R}}_+)\) onto \(L^2_\nu ({\mathbb {R}}_+)\). The inverse transform is given by
where the integral is convergent in \(L^2_\mu ({\mathbb {R}}_+)\).
Since J maps \(L^1_\mu ({\mathbb {R}}_+)\) into \(L^\infty ({\mathbb {R}}_+)\), see [18, (3.2)], Riesz–Thorin theorem ensures that J maps \(L^p_\mu ({\mathbb {R}}_+)\) to \(L^{p'}_\nu ({\mathbb {R}}_+)\) continuously, where \(1\le p\le 2\).
As the range of the Jacobi transform is different from its domain, introduction of a dual translation is necessary, see [19]. It is given by the next formula
where the dual kernel is
see [19, (4.14)]. That is the kernel is symmetric in its three variables. Moreover
see [19, Theorem 4.4 and (4.17)]. This ensures, in standard way, that
Subsequently, we need the next properties (see also [21, Section 2]).
Lemma 4.3
Let \(f\in C_0({\mathbb {R}}_+)\). Then
and
Proof
By [18, (4.2)]
Thus
where by the assumption on f Fubini theorem can be applied and the last integral is convergent. It proves (4.4).
Similarly, in view of [19, (4.16)]
which implies that
which, with the remark above, ensures (4.5).\(\square \)
Before stating the next theorem, we introduce the corresponding convolution. For appropriate functions
and
furthermore, the convolution is commutative and associative; see [18, (5.3), (5.4)].
Theorem 4.4
Let \(1\le p \le 2\), \(K\subset L^p_\mu ({\mathbb {R}}_+)\) a bounded set. Let \(\hat{K}:={J}(K)\).
Let us consider Example 2.8.
If K satisfies \(\mathbf{P_a}\), then \(\hat{K}\) fulfils \(\mathbf{P_{b_1}}\) and \(\mathbf{P_{b_2}}\).
If K satisfies \(\mathbf{P_{b_1}}\), then \(\hat{K}\) fulfils \(\mathbf{P_a}\).
Proof
Let \(f\in K\). In view of (4.5)
According to [17, Theorem 2, (ib)], if \(\eta \in {\mathbb {R}}\)
Thus, since \(\varrho >0\), \(\Vert T_d^{\eta +h}\hat{f}(\lambda )-T_d^\eta \hat{f}(\lambda )\Vert _{p',\nu }\le c(\varrho )h\Vert f\Vert _{p,\mu }\), so \(\mathbf{P_{b_1}}\) is satisfied for \(\hat{K}\).
Repeating the standard arguments, for an arbitrary \(\delta >0\), take a function \(g_\delta \), such that \({\textrm{supp}}g_\delta \subset [-\delta ,\delta ]\), \(\int _{{\mathbb {R}}_+}g_\delta (t)\textrm{d}\mu (t)=1\), \(\hat{g}_\delta >0\), cf. [19, Lemma 4.2]. As \(\hat{g}_\delta (\lambda )\) tends to zero when \(\lambda \) tends to infinity, choose R so large that \(\hat{g}_\delta (\lambda )<\frac{1}{2}\) if \(\lambda >R\). Then
5 Modulus of smoothness and K-functional: regularity with respect to an integral transform II
As it is well known, the modulus of smoothness generated by the standard translation is equivalent with the Peetre’s K-functional; see, e.g., [14, page 171]. This property is extended to Bessel translation; see [37]. Below, we derive the same equivalence for heat-diffusion semigroup, where the semigroup property implies arguments very similar to the standard case, and for Example 2.5, where the corresponding integral transform proves to be the right tool.
Recalling the notation of (3.5), we can define the Sobolev space generated by \(D_{(2)}\) as follows:
Then, the corresponding K-functional is
The moduli of smoothness generated by general translations are defined as follows.
Definition 5.1
Let
Let \(1\le p\le \infty \). The p-modulus of smoothness is
Remark 5.2
where \((T^t)^k=T^t\circ \dots \circ T^t\). For the standard and the heat-diffusion translation semigroups, \((T^t)^k\) becomes \(T^{kt}\).
Theorem 5.3
There are positive constants \(M=M(r,p)\) independent of f, such that for Examples 2.3 and 2.5
\(r\in {\mathbb {N}}\), \(1\le p\le \infty \), \(c={\textrm{ord}}(D_{(1)})\).
Remark 5.4
In our parabolic cases \(c=1\), in Bessel case \(c=2\); cf. [37].
Proof
By standard arguments
In view of (3.6) and Remark 5.2
The second estimation needs different methods in the two cases.
First, we deal with the heat semigroup. Let us introduce the next notation
Thus, according to (2.6)
Considering the operator norm, (5.2) and (5.3) \(g_{r,t} \in W^p_{r,L}\). By Minkowski’s inequality
Recalling the definition of \(g_{r,t}\), as above
To prove the second inequality for Example 2.5, we apply the integral transform method again. According to (5.1) and (4.1)
Let \(I_1:=I\cap [-1,1]\) and \(I_2:=I{\setminus } (-2,2)\). Let us define
such that \(0\le \eta \le 1\). Let \(\varepsilon >0\) and define \(\eta _\varepsilon (x):=\eta \left( \frac{x}{\varepsilon }\right) \). Let
It is enough to take an \(f\in L^1\cap L^p\). Define
Let \(0<t<\frac{1}{b^2\varepsilon ^2}\). Then, we have
Let
Then
Since
we have
We proceed similarly to estimate the first term of \(K_r(f,t^{r})_p\)
Let
We decompose
that is
Let \(\varepsilon =\frac{1}{b\sqrt{t}}\) and
Then
If N is large enough, we have
Turning our attention to the first term, we have
Since \(T^t\) is bounded
Thus, according to (5.5) and (5.6)
Recalling (5.4), the second inequality is proved.\(\square \)
Remark 5.5
This integral transform method works, because considering the two expressions \({\mathcal {I}}(\Delta _tf)=(1-\psi (t,\cdot ))\hat{f}\) and \({\mathcal {I}}D_{(2)}f=h(\cdot )\hat{f}\), we have
around zero. This property is the second regularity property of translation with respect to the integral transform. This is the situation, e.g., in Bessel case; see [37]. For instance, Example 2.4 does not possess second regularity property with respect to the cosine transform.
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This work was supported by the NKFIH-OTKA under Grant Nos. K128922 and K132097.
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Horváth, Á.P. Translation beyond Delsarte. Adv. Oper. Theory 8, 60 (2023). https://doi.org/10.1007/s43036-023-00287-5
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DOI: https://doi.org/10.1007/s43036-023-00287-5