Convergence Estimates for Stationary Radial Basis Function Interpolation and for Semi-discrete Collocation-Schemes

Abstract

We give a Fourier-theoretic analysis of the convergence of semi-discrete Radial Basis Function interpolation on regular grids and of the associated semi-discrete collocation schemes for evolution equations whose generator is a constant-coefficient pseudo-differential operator. We examine convergence in Wiener norm for a general class of basis functions which generalizes a class introduced earlier by M. Buhmann. We also discuss approximate approximation properties, in the sense of V. Maz’ya and G. Schmidt, both for interpolation and for the collocation scheme. Despite the use of the Wiener norm for the error we can allow a broad class of polynomially increasing functions. Our results apply to parabolic equations such as the heat equation, but also to non-local equations such as the Kolmogorov equation of a multi-dimensional Lévy-process, and to hyperbolic equations such as the (half-)wave equation or the transport equation.

Appendices

Appendix

Appendix A: Proof of Theorem 2.3

We prove the existence and main properties of the cardinal function associated to a basis function $\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n )$ as stated in Theorem 2.3. As mentioned, this was already done by Buhmann [4, 5] for a more restricted class of radial basis functions. The main difference in our treatment and that of [4] is the use of Lemma A.2 below, relating the decay at infinity of the Fourier transform of a function with its behavior in 0, which allows one to go beyond the class of basis functions considered by Buhmann.

It may be interesting to observe that the estimates (4) are similar to conditions (63) on symbols of pseudodifferential operators, except that the latter restrict the behavior at infinity instead of that at 0. Indeed, if (4) were required for all orders $\alpha$ (with constants depending on $\alpha$), then $(1 - \chi (\xi ) ) \widehat{\varphi } (\xi /|\xi |^2 ) \in S^{\kappa } _1 ({\mathbb {R}}^n )$ if $\chi \in C_c ^{\infty }({\mathbb {R}}^n )$ is equal to 1 on a neighborhood of 0. From this point of view, (5) corresponds to having an elliptic symbol, whence our terminology.

As a preliminary remark, we note that conditions (ii) and (iii) of Definition 2.1 imply that

$$\begin{aligned} |\partial _{\eta } ^{\alpha } (\widehat{\varphi }^{ \, -1 } ) | \le C |\eta |^{\kappa - |\alpha | } , \ \ |\eta | \le 1 , |\alpha | \le n + \lfloor \kappa \rfloor + 1 , \end{aligned}$$

(85)

as an easy proof by induction shows.

Turning to the proof of theorem 2.3, following Buhmann [4, 5], we start by defining $L_1$ as the inverse Fourier transform of the right hand side of (9), observing that since, by condition (iv) of Definition 2.1, the latter is an integrable function, $L_1$ is a well-defined continuous function. We first show that $L_1 (x)$ has the proper decay at infinity.

Theorem A.1

Let $\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n )$, and let

$$\begin{aligned} L_1 := \mathcal {F }^{-1 } \left( \frac{\widehat{\varphi } (\cdot ) }{ \sum _{k \in {\mathbb {Z}}^n } \widehat{\varphi } (\cdot + 2 \pi k ) } \right) . \end{aligned}$$

(86)

Then there exists a positive constant C such that

$$\begin{aligned} |L_1 (y) | \le C (1 + |y| )^{-\kappa - n } , \ \ y \in {\mathbb {R}}^n . \end{aligned}$$

(87)

The proof will use the following lemma, which basically is a special case of a classical estimate for kernels of convolution operators: see Stein [35], Proposition 2 of Chapter VI, Sect. 4.4.

Lemma A.2

Let $p > - n$ and let $a \in C^{\lfloor p \rfloor + n + 1 } ({\mathbb {R}}^n \setminus 0 )$ be supported in some ball B(0, R) such that³

$$\begin{aligned} |\partial _{\xi } ^{\alpha } a(\xi ) | \le C |\xi |^{p - |\alpha | } , \ \ , |\xi | \le R , \ |\alpha | \le \lfloor p \rfloor + n + 1 . \end{aligned}$$

(88)

Then the inverse Fourier transform $k = \mathcal {F }^{-1 } (a)$ satisfies

$$\begin{aligned} |k(x) | \le C_1 (1 + |x|)^{ - p - n } , \ \ x \in \mathbb {R}^n , \end{aligned}$$

(89)

with a constant $C_1 \le c_n C$, where $c_n$ only depends on n.

Stein in fact shows that if (88) is satisfied at all orders, without a necessarily being compactly supported, then k can be identified with a $C^{\infty }$-function away from 0, satisfying $|\partial _x ^{\alpha } k (x) | \le C_{\alpha } |x |^{ - p - n - |\alpha | }$ for all $\alpha$ and all x. This result is stated and proven there for $p = 0$, but the proof generalizes to any $p > -n .$ We only need this estimate for k(x) itself, in which case we only need (88) for the limited number of derivatives of a indicated, and we furthermore only need it for large |x| (note that if a has compact support, k is continuous, even $C^{\infty }$, and Stein‘s estimate for k(x) at 0 becomes trivial). The proof in [35] uses the Paley–Littlewood decomposition. An elementary prove of Lemma A.2 can be given by writing

$$\begin{aligned} k(x) = (2 \pi )^{-n } \int \limits _{{\mathbb {R}}^n } \, \chi (|x| \xi ) a(\xi ) e^{i (x , \xi ) } \, d\xi + (2 \pi )^{-n } \int \limits _{{\mathbb {R}}^n } \, (1 - \chi (|x| \xi ) ) a(\xi ) e^{i(x , \xi ) } \, d\xi . \end{aligned}$$

where $\chi \in C^{\infty }({\mathbb {R}}^n )$ with bounded derivatives such that $\chi (\xi ) = 0$ for $|\xi | \le 1$, $\chi (\xi ) = 1$ for $|\xi | \ge 2$, and integrating the first integral by parts $\lfloor p \rfloor + n + 1$ times.

Proof of theorem A.1

Let $\chi _0 \in C^{\infty } _c ({\mathbb {R}}^n )$ such that $\chi _0 (\eta ) = 1$ in a neighbourhood of 0 and $\mathrm{supp }(\chi _0 ) \subset (- \pi , \pi ) ^k .$ For $k \in {\mathbb {Z}}^n$, define $\chi _k$ by $\chi _k (\eta ) := \chi _0(\eta + 2 \pi k )$ and note that the supports of the $\chi _k$ are disjoint. Finally, let $\chi _c := 1 - \sum _k \chi _k$ ("c" for "complement"), so that $\chi _c$ together with the $\chi _k$’s form a partition of unit. Then

$$\begin{aligned} L_1 (x) = \ell _c (x) + \sum _{k \in {\mathbb {Z}}^n } \ell _k (x) , \end{aligned}$$

(90)

where

$$\begin{aligned} \ell _k = \mathcal {F }^{-1 } \left( \chi _k (\eta ) \frac{\widehat{\varphi } (\eta ) }{\sum _{\nu } \widehat{\varphi }(\eta + 2 \pi \nu ) } \right) , \ \ k \in {\mathbb {Z}}^n \ \text{ or } k = c . \end{aligned}$$

(91)

We examine the decay in x of each term separately.

Decay of $\ell _c$ The function $\chi _c (\eta ) / \sum _k \widehat{\varphi } (\eta + 2 \pi k )$ is in $C_b ^{\lfloor \kappa \rfloor + n + 1 } ({\mathbb {R}}^n )$, since the denominator is a strictly positive periodic function which is $C_b ^{\lfloor \kappa \rfloor + n + 1 }$ on the complement of $(2 \pi {\mathbb {Z}} )^n$ and therefore on the support of $\chi _c .$ Multiplying with $\widehat{\varphi }$, we find that $\widehat{\ell }_c (\eta )$ is $C^{\lfloor \kappa \rfloor + n + 1 }$ with integrable derivatives of all orders, which implies by the usual integration by parts argument that $|\ell _c (x) | \le C (1 + |x| )^{-(\lfloor \kappa \rfloor + n + 1 ) } \le C (1 + |x| )^{- \kappa - n } .$

Note that $\widehat{L }_1 (\eta )$ is at best $C^{\lfloor \kappa \rfloor }$ in the points of $2 \pi {\mathbb {Z}}^n$, so integration by parts will not give the required decay for of $\ell _k$, $k \in {\mathbb {Z}}^n .$ We use Lemma A.2 instead.

Decay of $\ell _0$ Since

$$\begin{aligned} \widehat{\ell }_0 (\eta ) - \chi _0 (\eta ) = \chi _0 (\eta ) \left( \widehat{L }_1 (\eta ) - 1 \right) = - \chi _0 (\eta ) \left( \frac{ \widehat{\varphi } (\eta ) ^{-1 } \sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) }{1 + \widehat{\varphi }(\eta ) ^{-1 } \sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k ) } \right) \end{aligned}$$

and since $\sum _{k \ne 0 } \widehat{\varphi } (\eta + 2 \pi k )$ is $C^{\lfloor \kappa \rfloor + n + 1 }$ on the suport of $\chi _0$, the estimates (85) easily imply that $\widehat{\ell }_0 (\eta ) - \chi _0 (\eta )$ satisfies condition (88) of Lemma A.2 with $p = \kappa$ (for $\psi := \widehat{\varphi } ^{-1 } \sum _{k \ne 0 } \widehat{\varphi } (\cdot + 2 \pi k )$ does, and then also $\psi / (1 + \psi )$). It follows that $|\ell _0 (x) | \le C (1 + |x |)^{- \kappa - n }$, since $\mathcal {F }^{-1 } (\chi _0 )$ is rapidly decreasing.

Decay of $\ell _k$, $k \ne 0, c$ This is similar, except that we have to pay attention to the size of the constant in front of the $(1 + |x| )^{ - \kappa - n } .$ The Fourier transform $\widehat{\ell }_k (\eta )$ will now be supported near $\eta = - 2 \pi k .$ Shifting by $2 \pi k$, we see that

$$\begin{aligned} \widehat{\ell }_k (\eta - 2 \pi k ) = \chi _0 (\eta ) \varphi (\eta - 2 \pi k ) \frac{\widehat{\varphi } (\eta )^{-1 } }{1 + \sum _{\nu \ne 0 } \widehat{\varphi } (\eta )^{-1 } \widehat{\varphi } (\eta + 2 \pi \nu ) } \end{aligned}$$

is supported in a small neighbourhood of 0, with derivatives of order $|\alpha | \le \lfloor \kappa \rfloor + n + 1$ bounded by $C (1 + |k | )^{-N } |\eta |^{\kappa - |\alpha | }$, with C independent of k. Lemma A.2 then implies that

$$\begin{aligned} |\ell _k (x) | = \left| \ell _k (x) e^{2 \pi i (k, x ) } \right| \le C (1 + |k| )^{-N } (1 + |x |)^{- \kappa - n } . \end{aligned}$$

Since $N > n$, summation over $k \in {\mathbb {Z}}^n$ completes the proof. $\square$

Remark A.3

The same arguments prove Theorem 6.1: if $a \in S^q _0 ({\mathbb {R}}^n )$ we have $\widehat{a(D)(L_1)} = a \widehat{\ell }_c + \sum _k a \widehat{\ell _k }$, and $a \widehat{\ell }_c$ has $\lceil \kappa \rceil + n + 1$ derivatives which are integrable since $N - q > n$, as has $a \chi _0 .$ Furthermore, $a \chi _0 (\widehat{L }_1 - 1 )$ and $a \chi _k \widehat{L }_1$, $k \ne 0$ satisfy the hypothesis (88) of Lemma A.2, the latter with constants bounded by $C |k |^{-(N - q ) }$ which are summable since $N > n + q .$ If $a(\xi ) = \xi ^{\alpha }$ we obtain theorem 2.7.

Finally, for symbols a in $\mathring{S }^q _1 ({\mathbb {R}}^n \setminus 0 )$ the estimate for $\mathcal {F }^{-1 } (\chi _0 a \widehat{L }_1 ) = \mathcal {F }^{-1 } ( a \chi _0 ) + \mathcal {F }^{-1 } ( a \chi _0 (\widehat{L }_1 - 1 ) )$ changes: Lemma A.2 shows that the first term decays like $(1 + |x | )^{- q - n }$ while the second one decays as $(1 + |x | )^{- q - \kappa - n } .$ The other terms in the decomposition of $\mathcal {F }^{-1 } (a \widehat{L }_1 )$ can be estimated by a constant times $(1 + |x )^{- \kappa - n }$, as before.

We continue with the proof of theorem 2.3. The expressions for $\widehat{\ell }_k$ above also show that $\widehat{L }_1$ satisfies the Strang-Fix conditions (10). Once we have defined $L_1$ through its Fourier transform, it is immediate to check that $L_1 (k) = \delta _{0k }$ for $k \in {\mathbb {Z}}^n$: indeed, by the $2 \pi$-periodicity of the denominator, writing the integral over ${\mathbb {R}}^n$ as a sum of integrals over translates of $(-\pi , \pi )^n$,

$$\begin{aligned} L_1 (k)= & {} \int \limits _{{\mathbb {R}}^n } \frac{\widehat{\varphi }(\eta ) }{\sum _{\nu } \widehat{\varphi }(\eta + 2 \pi \nu ) } e^{i k \eta } \frac{d \eta }{(2 \pi )^n } \\= & {} \int \limits _{(- \pi , \pi )^n } \frac{\sum _{\nu ' } \widehat{\varphi } (\eta + 2 \pi \nu ' ) }{\sum _{\nu } \widehat{\varphi }(\eta + 2 \pi \nu ) } e^{i k \eta } \frac{d \eta }{(2 \pi )^n } \\= & {} \int \limits _{(- \pi , \pi )^n } e^{i k \eta } \frac{d \eta }{(2 \pi )^n } \\= & {} \delta _{0k } . \end{aligned}$$

It remains to recognise $L_1$ as a sum of translates of $\varphi .$ To show this we first write the denominator of (9) as a Fourier series:

$$\begin{aligned} \left( \sum _k \widehat{\varphi }(\eta + 2 \pi k ) \right) ^{-1 } = \sum _k c_k e^{i k \eta } . \end{aligned}$$

(92)

One verifies by the similar arguments as the ones of the proof of theorem A.1 that

$$\begin{aligned} |c_k | \le C (1 + |k |)^{- \kappa - n } : \end{aligned}$$

(93)

write

$$\begin{aligned} c_k \!=\! (2 \pi )^{-n } \int \limits _{(- \pi , \pi )^n } \frac{\chi _0 (\eta ) }{\sum _{\nu } \widehat{\varphi } (\eta + 2 \pi \nu ) } e^{i (\eta , k ) } d\eta \!+\! (2 \pi )^{-n } \int \limits _{(- \pi , \pi )^n } \frac{1 - \chi _0 (\eta ) }{\sum _{\nu } \widehat{\varphi } (\eta + 2 \pi \nu ) } e^{i (\eta , k ) } d\eta , \end{aligned}$$

and estimate the first integral using Lemma A.2 and the second by integrating by parts.

It follows that (92) converges absolutely. We then claim that

$$\begin{aligned} L_1 (x) = \sum _k c_k \varphi (x - k ) , \end{aligned}$$

(94)

where the series converges absolutely and uniformly on compacta, by (93), since $\varphi (x)$ grows at most as $(1 + |x \ )^{\kappa - \varepsilon }$, by assumption. Formally, (94) follows by writing

$$\begin{aligned} L_1 (x) = \int \limits _{{\mathbb {R}}^n } \left( \sum _k c_k e^{i k \eta } \right) \widehat{\varphi } (\eta ) e^{i \eta x } \frac{d \eta }{(2 \pi )^n } = \sum _k c_k \varphi (x + k ) , \end{aligned}$$

except that the final step does not make sense for an arbitrary $\varphi \in \mathfrak {B }_{\kappa , N } ({\mathbb {R}}^n )$ since $\widehat{\varphi } (\eta )$ will not be integrable in 0 if $\kappa > n .$ We have to carefully distinguish between the tempered distribution $\widehat{\varphi }$ and the locally integrable function $\eta \rightarrow \widehat{\varphi } (\eta )$ with which it can be identified on ${\mathbb {R}}^n \setminus 0 .$ The relation between the two is given by the following identity: there exist constants $c_{\alpha }$, $|\alpha | \le \lceil \kappa \rceil - 1$ such that for all $\psi \in \mathcal {S }({\mathbb {R}}^n )$,

$$\begin{aligned} \langle \widehat{\varphi } , \psi \rangle= & {} \int \limits _{|\eta | \le 1 } \, \widehat{\varphi } (\eta ) \left( \psi (\eta ) - \sum _{|\alpha | \le \lfloor \kappa \rfloor - n } \frac{\psi ^{(\alpha ) }(0) }{\alpha ! } \eta ^{\alpha } \right) \, d\eta \nonumber \\&+ \int \limits _{|\eta | \ge 1 } \widehat{\varphi } (\eta ) \psi (\eta ) \, d\eta \nonumber \\&+ \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } (-1 )^{|\alpha | } c_{\alpha } \psi ^{(\alpha ) }(0) , \end{aligned}$$

(95)

where the first sum is empty if $\kappa < n .$ Indeed, the first integral converges since $|\eta |^{\lfloor \kappa \rfloor - n + 1 } \widehat{\varphi } (\eta )$ has an integrable singularity at 0. The sum of the two integrals on the right defines a tempered distribution. If we denote this distribution by $\Lambda _{\widehat{\varphi } }$ then the restriction of $\Lambda _{\widehat{\varphi } }$ to ${\mathbb {R}}^n \setminus 0$ can be identified with the function $\widehat{\varphi } (\eta ) .$ The difference $\widehat{\varphi } - \Lambda _{\widehat{\varphi } }$ is then supported in 0, and therefore a linear combination $\sum _{|\alpha | \le p } c_{\alpha } \delta _0 ^{(\alpha ) }$ of derivatives of the delta distribution in 0. To bound p, we use the following lemma, whose proof we postpone till the end of his section:

Lemma A.4

The inverse Fourier transform $\mathcal {F }^{-1 } \left( \Lambda _{\widehat{\varphi } } \right)$ is a continuous function which is bounded if $\kappa < n$ while if $\kappa \ge n$ it is bounded by $C (|x |^{\kappa - n } +1 )$ if $\kappa \notin {\mathbb {N}}$ and by $C (|x |^{\kappa - n } \log |x | + 1 )$ if $\kappa - n \in {\mathbb {N}} .$

Since the inverse Fourier transform of $\sum _{|\alpha | \le p } c_{\alpha } \delta _0 ^{(\alpha ) }$ is a polynomial of order p, and since, by assumption, $\varphi (x)$ has polynomial growth of order strictly less than $\kappa$, it follows that $p < \kappa$, which is equivalent to $p \le \lceil \kappa \rceil - 1 .$

We now use (95) to prove that (94) holds as tempered distributions, that is, if $\psi \in \mathcal {S }({\mathbb {R}}^n )$, then

$$\begin{aligned} \langle L_1 , \widehat{\psi } \rangle = \left\langle \sum _k c_k \varphi ( \cdot + k ) , \widehat{\psi } \right\rangle . \end{aligned}$$

(96)

If we let

$$\begin{aligned} \Psi (\eta ) := \frac{\psi (\eta ) }{\sum _k \widehat{\varphi } (\eta + 2 \pi k ) } . \end{aligned}$$

then $\Psi$ is $C^{\lfloor \kappa \rfloor }$ if $\kappa \notin {\mathbb {N}}$, and $C^{\kappa - 1 , 1 }$ if $\kappa \in {\mathbb {N}}^*$, with all its derivatives rapidly decreasing. To obtain a function in the Schwartz class $\mathcal {S }({\mathbb {R}}^n )$ we convolve with $\chi _{\varepsilon } (x) := \varepsilon ^{-n } \chi (x/\varepsilon )$, where $\chi \in C^{\infty }_c ({\mathbb {R}}^n )$ with integral 1. Let $\Psi _{\varepsilon } := \chi _{\varepsilon } * \Psi .$ Then we first claim that

$$\begin{aligned} \langle \widehat{\varphi } , \Psi _{\varepsilon } \rangle \rightarrow \int \limits _{{\mathbb {R}}^n } \, \widehat{L }_1 (\eta ) \psi (\eta ) \, d\eta . \end{aligned}$$

(97)

To show this it suffices to consider the case that $\kappa \ge n$, since $\widehat{\varphi }$ is integrable if $\kappa < n$ and $\Psi _{\varepsilon }$ converges uniformly. If we assume for example that $\mathrm{supp }(\chi ) \subset B(0, 1 )$ then by the Taylor expansion with remainder there exists a constant $C > 0$ such that for all $\varepsilon \le 1$,

$$\begin{aligned} \left| \Psi _{\varepsilon } (\eta ) - \sum _{|\alpha | \le \lfloor \kappa \rfloor - n } \frac{ \Psi _{\varepsilon } ^{(\alpha ) } (0) }{\alpha ! } \eta ^{\alpha } \right|\le & {} C \max _{|\beta | = \lfloor \kappa \rfloor - n + 1 } \sup _{B(0, 1 ) } | \Psi _{\varepsilon } ^{(\beta ) } | \cdot |\eta |^{\lfloor \kappa \rfloor - n + 1 }\\\le & {} C \max _{|\beta | = \lfloor \kappa \rfloor - n + 1 } \sup _{B(0, 2 ) } |\Psi ^{(\beta ) } | \cdot |\eta |^{\lfloor \kappa \rfloor - n + 1 } , \end{aligned}$$

where we note that if $n \ge 2$ or if $\kappa \notin {\mathbb {N}}^*$, then derivatives of $\Psi$ of order $\lfloor \kappa \rfloor - n + 1$ exist, while if $n = 1$ and $\kappa \in {\mathbb {N}}^*$, these derivatives exist a.e. but are uniformly bounded, and the estimate remains true. Next, $\Psi ^{(\alpha ) } _{\varepsilon } (x) \rightarrow \Psi ^{(\alpha ) } (x)$ for $|\alpha | \le \lceil \kappa \rceil - 1$ since $\Psi$ is $C^{\lceil \kappa \rceil - 1 } .$ Furthermore, since $(\sum _k \widehat{\varphi } (\eta + 2 \pi k ) )^{-1 }$ vanishes of order $\kappa$ in 0, it follows that $\Psi ^{(\alpha ) } (0) = 0$ for $|\alpha | \le \lceil \kappa \rceil - 1$ and (97) follows by dominated convergence.

Since $\Psi _{\varepsilon }$ is Schwartz-class, we have $\langle \widehat{\varphi } , \Psi _{\varepsilon } \rangle = \langle \varphi , \widehat{\Psi }_{\varepsilon } \rangle .$ By (92) $\Psi = \left( \sum _k c_k e^{-i (k, \eta ) } \right) \psi (\eta )$, which can be interpreted as the product of a tempered distribution and a test function, whose Fourier transform equals

$$\begin{aligned} \widehat{\Psi } (x) = \sum _k c_k \widehat{\psi } (x - k ) . \end{aligned}$$

One easily verifies using (93) that $|\widehat{\Psi } (x) | \le C (1 + |x| )^{- \kappa - n } .$

Since $\widehat{\Psi }_{\varepsilon } (x) = \widehat{\chi } (\varepsilon x ) \widehat{\Psi } (x)$, and since $|\varphi (x) | \ \le C (1 + |x | )^{\kappa - \rho }$ for some $\rho > 0$, Lebesgue’s dominated convergence theorem then shows that

$$\begin{aligned} \langle \varphi , \widehat{\Psi }_{\varepsilon } \rangle= & {} \int \limits _{{\mathbb {R}}^n } \, \varphi (x) \widehat{\Psi } (x) \widehat{\chi }(\varepsilon x ) \, dx \\\rightarrow & {} \int \limits _{{\mathbb {R}}^n } \varphi (x) \left( \sum _k c_k \widehat{\psi } (x - k ) \right) \, dx . \end{aligned}$$

Finally, one checks that the functions $(x, k ) \rightarrow c_k \varphi (x) \widehat{\psi } (x - k )$ and $(x, k ) \rightarrow c_k \varphi (x + k ) \widehat{\psi } (x)$ are integrable on ${\mathbb {R}}^n \times {\mathbb {Z}}^n$ with respect to the product of the Lebesgue measure and the counting measure. A double application of Fubini’s theorem then shows that the right hand side equals

$$\begin{aligned} \int \limits _{{\mathbb {R}}^n } \left( \sum _k c_k \varphi (x + k ) \right) \widehat{\psi }(x) \, dx , \end{aligned}$$

which proves (96). The pointwise identity (94) follows since both sides are continuous functions.

Proof of Lemma A.4

We give a proof for convenience of the reader. If $\kappa < n$, $\widehat{\varphi }$ is integrable, and its inverse Fourier transform is a bounded continuous function, so suppose that $\kappa \ge n .$ Since $\mathbf {1 }_{\{ |\eta | \ge 1 \} } \widehat{\varphi } (\eta )$ is integrable, its inverse Fourier transform is a bounded continuous function, and it therefore suffices to examine the inverse Fourier transform of the tempered distribution defined by the first integral on the right hand side of (95). This distribution being of compact support, its inverse Fourier transform is the function k(x) obtained by taking $\psi (\eta ) = (2\pi )^{-n } e^{i(x, \eta ) }$:

$$\begin{aligned} k(x) := (2 \pi )^{-n } \int \limits _{|\eta | \le 1 } \, \widehat{\varphi } (\eta ) \left( e^{i(x, \eta ) } - \sum _{j \le \nu } \frac{i^j (x, \eta ) ^j }{j! } \right) \, d\eta , \end{aligned}$$

(98)

where $\nu := \lfloor \kappa \rfloor - n .$ This is a $C^{\infty }$-function which can be bounded by

$$\begin{aligned} |k(x) |\le & {} C \int \limits _{|\eta | \le 1 } \, |\eta |^{- \kappa } \left| e^{i(x, \eta ) } - \sum _{j \le \nu } \frac{i^j (x, \eta ) ^j }{j! } \right| \, d\eta \\= & {} C |x| ^{\kappa - n } \int \limits _{|\eta | \le |x| } |\eta |^{- \kappa } \left| e^{i(\tfrac{x }{|x | } , \eta ) } - \sum _{j \le \nu } \frac{i^j (\tfrac{x }{|x | } , \eta ) ^j }{j! } \right| \, d\eta . \end{aligned}$$

Split the integral into an integral over $|\eta | \le c$ and one over the complement, where $c > 0$ is some fixed number and where we assume wlog that $|x | > c .$ The first integral converges absolutely, since

$$\begin{aligned} \left| e^{i(\tfrac{x }{|x | } , \eta ) } - \sum _{j \le \nu }\frac{i^j (\tfrac{x }{|x | } , \eta ) ^j }{j! } \right| \le \frac{1 }{\nu ! } | (\tfrac{x }{|x | } , \eta ) |^{\nu + 1 } \le \frac{|\eta |^{\nu + 1 } }{\nu ! } , \end{aligned}$$

and $\nu - \kappa + n = \lfloor \kappa \rfloor - \kappa > -1$, and we can bound its contribution to k(x) by $C |x |^{\kappa - n } .$ If $\kappa \notin {\mathbb {N}}$ the integral over $| \eta | > c$ can be bounded by a constant times

$$\begin{aligned} |x |^{\kappa - n } \sum _{j = 0 } ^{\nu } \int \limits _c ^{|x | } r^{- \kappa + j + n - 1 } dr = |x |^{\kappa - n } \sum _{j = 0 } ^{\nu } \frac{1 }{j - \kappa + n } \left( |x|^{j - \kappa + n } - c^{j - \kappa + n } \right) , \end{aligned}$$

which is bounded by $C (|x |^{\kappa - n} + 1 )$ since $\nu - \kappa + n < 0$. Finally, if $n \le \kappa \in {\mathbb {N}}$, then $\nu = \kappa - n$ and

$$\begin{aligned}&|x| ^{\kappa - n } \sum _{j = 0 } ^{\kappa - n } \int \limits _c ^{|x | } r^{j - (\kappa - n ) - 1 } dr \\&\quad = |x|^{\kappa - n } \sum _{j = 0 } ^{\kappa - n - 1 } \frac{1 }{j - \kappa + n } \left( |x| ^{j - (\kappa - n ) } - c^{j - (\kappa - n )} \right) + \log (|x | / c ) \\&\qquad \le C |x|^{\kappa - n } (\log |x | + 1 ) . \end{aligned}$$

$\square$

Remark A.5

More generally, if F is a measurable function such that $F(\eta ) = O(|\eta |^{- \kappa } )$ near 0 and $|\eta |^r F \cdot \mathbf {1 }_{ \{ |\eta | \ge 1 \} } \in L^1 ({\mathbb {R}}^n )$ for some $r \in {\mathbb {N}}$, then $K := \mathcal {F }^{-1 } (\Lambda _F )$, with $\Lambda _F$ defined by the first two terms of (95) with $\widehat{\varphi }$ replaced by F, is a $C^r$-function whose derivatives can be bounded by

$$\begin{aligned} |\partial _x ^{\alpha } K(x) | \le \left\{ \begin{array}{ll} C (|x |^{ \max ( \kappa - n - |\alpha | , \, 0 ) } + 1 ) , &{}\kappa \notin {\mathbb {N}}\ \text{ or } \kappa - n - |\alpha | < 0 \\ C (|x |^{ \max ( \kappa - n - |\alpha | , \, 0 ) } \log |x | + 1 ) , &{}\kappa - n - |\alpha | \in {\mathbb {N}}^* , \end{array} \right. \end{aligned}$$

where $|\alpha | \le r .$ This immediately implies symbol estimates for the generators of pure-jump Lévy processes ($\Sigma = 0$) with Lévy measure of the form

$$\begin{aligned} d\nu (\eta ) = \frac{h(\eta ) }{|\eta |^Y } d\eta , \ \ Y < n + 2 , \end{aligned}$$

where $h = h(\eta )$ is a rapidly decreasing function. Taking $\chi$ equal to the characteristic function of the unit ball in the Lévy–Khintchine formula (84), and setting $\mu = 0$ (without essential loss of generality), we see that the $\psi (\xi )$ is exactly the inverse Fourier transform of the distribution $\Lambda _{ |\eta |^{- Y } h(\eta ) }$, modulo a constant which equals $\int (1 - \chi ) d\nu (x) .$ This certainly implies that $\psi \in S^{\max (Y - n , \, 0 ) } _0$ if $Y \notin {\mathbb {N}}$ (there is an additional decay-factor of $|\xi |^{-1 }$ for the first derivative if $n< Y < n + 1$), while $\psi / \log (|\xi |^2 + 1 ) \in S_0 ^{Y - n }$ if $Y = n$ or $Y = n + 1 .$

Appendix B: Some Technical Proofs

1.1 Appendix B.1: Proof of Lemma 3.5

Let $F \in L^1 \left( {\mathbb {R}}^n \setminus 0 , (|\xi |^{\kappa } \wedge 1 ) d\xi \right)$, with $\kappa \ge 0 .$ Then F gives rise to a tempered distribution $\Lambda _F \in \mathcal {S }' ({\mathbb {R}}^n )$ defined as follows: if g is a bounded compactly supported function which is equal to 1 on a neighbourhood of 0, then

$$\begin{aligned} \langle \Lambda _F, \psi \rangle:= & {} \int \limits _{{\mathbb {R}}^n } \left( \psi (\xi ) - \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } \psi ^{(\alpha ) } (0) \frac{\xi ^{\alpha } }{\alpha ! } \right) g(\xi ) F (\xi ) \, d\xi \\&+ \int \limits _{{\mathbb {R}}^n } (1 - g(\xi ) ) F(\xi ) \psi (\xi ) \, d\xi , \ \ \psi \in \mathcal {S }({\mathbb {R}}^n ) \, , \nonumber \end{aligned}$$

(99)

cf. (95). The integral converges since $\psi - \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } \psi ^{(\alpha ) } (0) \xi ^{\alpha } / \alpha ! = O(|\xi |^{\lceil \kappa \rceil } ) = O(|\xi |^{\kappa } )$ in a neighbourhood of 0 and defines a distribution of order $\lceil \kappa \rceil - 1 .$ Note that $\Lambda _F$ coincides on ${\mathbb {R}}^n \setminus 0$ with the locally integrable function F.

We next observe that $\Lambda _F$ extends to a continuous linear functional on the Hölder space $C_b ^{\lceil \kappa \rceil - 1 , \lambda } := C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n )$ with $\lambda = \kappa - (\lceil \kappa \rceil - 1 ) .$ Indeed, if $\psi \in C^{K , \lambda } ({\mathbb {R}}^n )$, then the Taylor expansion formula with integral remainder term easily implies that

$$\begin{aligned} \left| \psi (\xi ) - \sum _{ |\alpha | \le K } \psi ^{(\alpha ) } (0) \xi ^{\alpha } / \alpha ! \right| \le C \left( \sum _{ |\beta | = K } || \psi ^{(\beta ) } ||_{0 , \lambda } \right) |\xi |^{K + \lambda } , \end{aligned}$$

(100)

which shows, with $K = \lceil \kappa \rceil - 1$ and $\lambda = \kappa - (\lceil \kappa \rceil - 1 )$, that $\langle \Lambda _F , \psi \rangle$ is well-defined and continuous.

We can, in particular, let $\Lambda _F$ act on the imaginary exponentials $\xi \rightarrow e^{i (x, \xi ) } .$ The function

$$\begin{aligned} \check{F } : x \rightarrow (2 \pi )^{-n } \left\langle \Lambda _F , e^{i (x, \xi ) } \right\rangle . \end{aligned}$$

is then found to be bounded by $C (1 + |x | )^{\kappa }$, since $|| e^{i(x, \xi ) } ||_{K , \lambda } \le C (1 + |x |^{K + \lambda } )$, and one easily verifies that the inverse Fourier transform of $\Lambda _F$ coincides with $\check{F } .$ If $\kappa \in {\mathbb {N}}$ one has the stronger estimate

$$\begin{aligned} |\check{F } (x) | = o(|x |^{\kappa } ) , \ \ |x | \rightarrow \infty , \end{aligned}$$

(101)

which can be seen as follows: write $F= \chi F + (1 - \chi ) F$ with $\chi$ the characteristic function of a small ball around 0. Since $(1 - \chi ) F$ is integrable, its inverse Fourier transform tends to 0 at infinity, by the Riemann-Lebesgue lemma. We can therefore wlog assume that F is supported in $\{ g = 1 \} .$ If we apply (99) with $\psi (\xi ) = e^{i (x, \xi ) }$ then⁴

$$\begin{aligned} \check{F } (x)= & {} \sum _{|\alpha |= \kappa } \int \limits _{{\mathbb {R}}^n } F(\xi ) \frac{(ix )^{\alpha } \xi ^{\alpha } }{\alpha ! } \left( \int \limits _0 ^1 \frac{ \ \ (1 - s )^{\kappa - 1 } }{(\kappa - 1 ) ! } e^{i s (x, \xi ) } ds \right) \frac{d\xi }{(2 \pi )^n } \\=: & {} \sum _{|\alpha | = \kappa } (i x )^{\alpha } \int \limits _0 ^1 \check{F }_{\alpha } (sx ) \frac{ \ \ (1 - s )^{\kappa - 1 } }{(\kappa - 1 ) ! } ds , \end{aligned}$$

where $\check{F }_{\alpha } (x)$ is the inverse Fourier transform of the $L^1$-function $\xi \rightarrow \xi ^{\alpha } F(\xi ) .$ By the Riemann-Lebesgue lemma, $\check{F }_{\alpha } (sx) \rightarrow 0$ as $x \rightarrow \infty$, for all $s \in (0, 1 ]$, and the same is true for the integral over $s \in [0, 1 ]$, by the dominated convergence theorem (the $\check{F }_{\alpha }$ are bounded). Hence $\check{F }(x) / |x |^{\kappa } \rightarrow 0$ for $x \rightarrow \infty$, as claimed.

Now let f be a measurable function on ${\mathbb {R}}^n$ of polynomial growth of order strictly less than $\kappa$, such that its Fourier transform $\widehat{f }$ (in the sense of tempered distributions) satisfies

$$\begin{aligned} \widehat{f } \big |_{{\mathbb {R}}^n \setminus 0 } \in L^1 \left( {\mathbb {R}}^n , (|\xi |^{\kappa } \wedge 1 ) d\xi \right) . \end{aligned}$$

We write $\Lambda _{\widehat{f } }$ for $\Lambda _{\widehat{f } |_{{\mathbb {R}}^n \setminus 0 } } .$ Then $\widehat{f } - \Lambda _{\widehat{f } }$ is a distribution which is supported in 0, and therefore of the form $\sum _{|\alpha | \le N } c_{\alpha } \delta _0 ^{(\alpha ) }$ for certain $N \in {\mathbb {N}}$ and $c_{\alpha } \in {\mathbb {C}}$ with $\sum _{|\alpha | = N } |c_{\alpha } | \ne 0 .$ Since the inverse Fourier transform of $\widehat{f } - \Lambda _{\widehat{f } }$ is a polynomial of degree N, it follows that $N \le \lceil \kappa \rceil - 1$, the largest integer which is strictly smaller than $\kappa$, since otherwise $|f(x) - \check{F}(x)|$ would grow at a rate of at least $|x |^{\lceil \kappa \rceil }$ in certain directions. If $\kappa \notin {\mathbb {N}}$ this would contradict the bound $\check{F }(x) = 0(|x |^{\kappa } )$, and if $\kappa \in {\mathbb {N}}$ this would contradict (101).

In follows that $\widehat{f } = \Lambda _{\widehat{f } } \, + \, \sum _{|\alpha | \le N } c_{\alpha } \delta _0 ^{(\alpha ) }$ also extends to a continuous linear functional on $C^{\lceil \kappa \rceil - 1 , \kappa - (\lceil \kappa \rceil - 1 ) } .$ We exploit this to define $\Sigma _h (\widehat{f } )$ by duality.

If $\psi \in \mathcal {S }({\mathbb {R}}^n )$, we let

$$\begin{aligned} \Sigma _h ' (\psi ) := \sum _k \psi (\xi + 2 \pi h^{-1 } k ) \, \widehat{L }_1 (h \xi + 2 \pi k ) . \end{aligned}$$

(102)

Note that $\Sigma _h '$ is the formal (real) adjoint of $\Sigma _h .$ By Lemma 2.5, $\Sigma _h ' (\psi )$ is $C_b ^{\lceil \kappa \rceil - 1 , \lambda }$ with $\lambda = \kappa - (\lceil \kappa \rceil - 1 )$ and uniformly bounded together with all its derivatives, since $2 \pi h^{-1 }$-periodic. In fact, this is true even if $\psi \in C_b ^{\lceil \kappa \rceil - 1 , \lambda }$ with the same $\lambda$, on account of the decay at infinity of $\widehat{L }_1 .$ We can then define $\Sigma _h (\widehat{f } )$, as a tempered distribution and, more generally, as a bounded linear functional on $C_b ^{\lceil \kappa \rceil - 1 , \lambda } ({\mathbb {R}}^n )$ by

$$\begin{aligned} \left\langle \Sigma _h (\widehat{f } ) , \psi \right\rangle := \left\langle \widehat{f } , \Sigma _h ' (\psi ) \right\rangle . \end{aligned}$$

(103)

We next check that $\Sigma _h (\widehat{f } )$ is the Fourier transform, in distribution sense, of $s _h [f ] .$ This is done by a standard approximation argument, with some care with the spaces in which the approximating sequence converges. We first note that we can assume without loss of generality that $\widehat{f }$ is compactly supported: indeed, we can write $f = f_1 + f_2$ with $\widehat{f }_1$ compactly supported and $\widehat{f }_2 \in L^1 ({\mathbb {R}}^n )$, and we know already that $\widehat{s _h [f_2 ] } = \Sigma _h (\widehat{f }_2 ) .$

So let $\widehat{f }$ be compactly supported, and let $\chi \in C^{\infty } _c ({\mathbb {R}}^n )$ be a non-negative symmetric function with $\int _{{\mathbb {R}}^n } \chi d\eta = 1 .$ Let $\chi _{\varepsilon } (\eta ) := \varepsilon ^{-n } \chi (\eta / \varepsilon ) .$ Then $\widehat{f }* \chi _{\varepsilon } \in C^{\infty } _c ({\mathbb {R}}^n ) .$

Lemma B.1

$\widehat{f } * \chi _{\varepsilon } \rightarrow \widehat{f }$ in the dual of $C^{K , \lambda }$, with $K = \lceil \kappa \rceil - 1$ and $\lambda = \kappa - K .$

Proof

On account of the symmetry of $\chi$,

$$\begin{aligned} \langle \widehat{f } * \chi _{\varepsilon } , \psi \rangle = \langle \widehat{f } , \psi * \chi _{\varepsilon } \rangle , \end{aligned}$$

which is valid both for Schwarz-class functions $\psi \in \mathcal {S }$ and for $\psi \in C^{K , \lambda } .$ Write $\psi _{\varepsilon } := \psi * \chi _{\varepsilon } .$ If $\psi \in C^{K , \lambda }$, then $\psi _{\varepsilon } ^{(\alpha ) } (x) \rightarrow \psi ^{(\alpha ) } (x)$ pointwise on ${\mathbb {R}}^n$ for all $|\alpha | \le K$, while a trivial estimate shows that $|| \psi _{\varepsilon } ^{(\alpha ) } ||_{0, \lambda } \le || \psi ^{(\alpha ) } ||_{0 , \lambda }$, uniformly in $\varepsilon > 0$, for $|\alpha | = K .$ This, together with the remainder estimate (100), the integrability of $\widehat{f } (\xi ) (|\xi |^{\kappa } \wedge 1 )$ and Lebesgue’s dominated convergence theorem, implies that $\langle \Lambda _{\widehat{f } } , \psi _{\varepsilon } \rangle \rightarrow \langle \Lambda _{\widehat{f } } , \psi \rangle .$ Since also $\langle \delta _0 ^{(\alpha ) } , \psi _{\varepsilon } \rangle \rightarrow \langle \delta ^{(\alpha ) } _0 , \psi \rangle$ for all $|\alpha | \le K$, the lemma follows. $\square$

The lemma immediately implies that if $\psi \in \mathcal {S } ({\mathbb {R}}^n )$, then $\langle \widehat{f } * \chi _{\varepsilon } , \Sigma _h ' (\psi ) \rangle \rightarrow \langle \widehat{f } , \Sigma _h ' (\psi ) \rangle$, so $\Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) \rightarrow \Sigma _h (\widehat{f } )$ in $\mathcal {S }' ({\mathbb {R}}^n )$ and even in $(C^{K , \lambda } ) '$ with K and $\lambda$ as above.

On the other hand, if we let $f_{\varepsilon }$ be the inverse Fourier transform of $\widehat{f } * \chi _{\varepsilon }$, then $f_{\varepsilon } \in \mathcal {S } ({\mathbb {R}}^n )$ since $\widehat{f } * \chi _{\varepsilon }$ is, and $\widehat{s_h [ f_{\varepsilon } ] } = \Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) .$ We have that $f_{\varepsilon } (x) = (2 \pi )^{n } f(x) \check{\chi } (\varepsilon x )$, with $\check{\chi }$ the inverse Fourier transform of $\chi$, so $\check{\chi } \in \mathcal {S }({\mathbb {R}}^n )$ and $(2 \pi )^n \check{\chi } (0) = 1 .$ By hypotheses, $f \in L^{\infty } _{- p }$ for some $p < \kappa .$ If $a > 0$ such that $p + a < \kappa$, then writing $\widetilde{\chi } := (2 \pi )^n \check{\chi }$,

$$\begin{aligned} || s_h [f_{\varepsilon } ] - s_h [f ] ||_{\infty , -(p + a ) }\le & {} C || f \left( \widetilde{\chi } (\varepsilon \cdot ) - 1 \right) ||_{\infty , -(p + a ) } \\\le & {} C || f ||_{\infty , -p } \, \sup _{x \in {\mathbb {R}}^n } \frac{ |\widetilde{\chi } (\varepsilon x ) - 1 | }{(1 + |x | )^a } \rightarrow 0 , \end{aligned}$$

as $\varepsilon \rightarrow 0$, using for example the first order Taylor expansion for $\widetilde{\chi }$ for $|x | \le \varepsilon ^{- 1 / 2 }$ plus a trivial estimate for $|x | > \varepsilon ^{-1 / 2 } .$ This certainly implies that $s_h [ f_{\varepsilon } ] \rightarrow s_h [f ]$ in $\mathcal {S }' ({\mathbb {R}}^n )$, so we conclude that $\widehat{s_h [f_{\varepsilon } ] } = \Sigma _h ( \widehat{f } * \chi _{\varepsilon } ) \rightarrow \widehat{s_h [f ] }$ and therefore $\widehat{s_h [f ] } = \Sigma _h (\widehat{f } )$ as distributions.

We finally show that $\Sigma _h (\widehat{f } ) = \widehat{f } + F$, where F is the $L^1$-function

$$\begin{aligned} F (\xi ) = \widehat{f }(\xi ) (\widehat{L }_1 (h \xi ) - 1 ) + \sum _{k \ne 0 } \widehat{f }(\xi + 2 \pi h^{-1 } k ) \widehat{L }_1 (h \xi ) . \end{aligned}$$

(104)

We first check that F is well-defined and in $L^1$: first of all, each of the terms on the right hand side is in $L^1$, on account of the Fix–Strang conditions satisfied by for $\widehat{L }_1$ and the integrability of $(|\xi |^{\kappa } \wedge 1 ) \widehat{f }(\xi ) .$ Next, the function $(\xi , k ) \rightarrow \widehat{f }(\xi + 2 \pi h^{-1 } k ) (\widehat{L }_1 (h \xi ) - \delta _{0k } )$ is absolutely integrable on ${\mathbb {R}}^n \times {\mathbb {Z}}^n$ with respect to the product of Lebesgue measure and the counting measure, since

$$\begin{aligned}&\sum _k \int \limits _{{\mathbb {R}}^n } | \widehat{f }(\xi + 2 \pi h^{-1 } k ) | \, |(\widehat{L }_1 (h \xi ) - \delta _{0k } ) | \, d\xi \\&\qquad = \int \limits _{{\mathbb {R}}^n } (1 - \widehat{L }_1 (h \xi ) ) |\widehat{f }(\xi ) | d\xi + \sum _{k \ne 0 } \widehat{L }_1 (h \xi + 2 \pi k ) |\widehat{f }(\xi ) | d\xi \\&\qquad = 2 \int \limits _{{\mathbb {R}}^n } (1 - \widehat{L }_1 (h \xi ) ) |\widehat{f } (\xi ) | d\xi . \end{aligned}$$

Fubini’s theorem then implies that $F(\xi )$ is well-defined for almost all $\xi \in {\mathbb {R}}^n$ and that $F \in L^1 ({\mathbb {R}}^n ) .$ If $\psi \in \mathcal {S } ({\mathbb {R}}^n )$, then a double application of Fubini will show that

$$\begin{aligned}&\int \limits _{{\mathbb {R}}^n } F(\xi ) \psi (\xi ) d\xi \\&\quad = \int \limits _{{\mathbb {R}}^n } \left( \psi (\xi ) (\widehat{L }_1 (h \xi ) - 1 ) + \sum _{k \ne 0 } \psi (\xi + 2 \pi h^{-1 } k ) \widehat{L }_1 (h \xi + 2 \pi k ) \right) \widehat{f }(\xi ) \, d\xi \\&\quad = \int \limits _{{\mathbb {R}}^n } (\Sigma _h ' (\psi ) - \psi ) \widehat{f }(\xi ) d\xi . \end{aligned}$$

Since, by the Fix–Strang conditions (10), all derivatives of order $\le \lceil \kappa \rceil - 1$ of $\Sigma _h (\psi ) - \psi$ in 0 are 0, the last integral is equal to $\langle \widehat{f } , \Sigma _h ' (\psi ) - \psi \rangle = \langle \Sigma _h (\widehat{f } )- \widehat{f } , \psi \rangle$, and therefore $\Sigma _h (\widehat{f } ) - \widehat{f } = F$, which finishes the proof of lemma 3.5.

Remark B.2

The lemma and its proof generalizes to f’s such that $\widehat{f } |_{{\mathbb {R}}^n \setminus 0 }$ is a finite Borel measure with respect to which the function $|\xi |^{\kappa } \wedge 1$ is integrable, provided that $\kappa \notin {\mathbb {N}}$ (the reason being that we then no longer have (101)).

1.2 Appendix B.2: Proof of Lemma 5.4

It again suffices to consider the case of compactly supported $\widehat{f }$’s. We use the notations of the proof of lemma 3.5 above: in particular, let $f_{\varepsilon }$ be the inverse Fourier transform of $\widehat{f } * \chi _{\varepsilon }$ where $\chi _{\varepsilon } = \varepsilon ^{-n } \chi ( \cdot / \varepsilon )$ is an approximation of the identity. We have seen that $\Sigma _h (\widehat{f }_{\varepsilon } ) \rightarrow \Sigma _h (\widehat{f } )$ in $\left( C^{K , \lambda } \right) '$, where $K = \lceil \kappa \rceil - 1$ and $\lambda = \kappa - K .$ Since $e^{- h^{-2 } t G (h \cdot ) } \in C^{K , \lambda }$ this implies that

$$\begin{aligned} e^{- h^{-2 } t G (h \cdot ) } \Sigma _h \left( \widehat{f }_{\varepsilon } \right) \rightarrow e^{- h^{-2 } t G (h \cdot ) } \Sigma _h (\widehat{f } ) \end{aligned}$$

in $\left( C^{K , \lambda } \right) '$ and hence in $\mathcal {S }' ({\mathbb {R}}^n ) .$

On the other hand, we have seen in the proof of lemma 3.5 that $f_{\varepsilon } \rightarrow f$ in $L^{\infty } _{- p - a }$ if $a > 0 .$ Hence by Lemma 5.1, if $a < \kappa - p$ then $u_h [f_{\varepsilon } ] \rightarrow u_h [f ]$ in $L^{\infty } _{- p - a }$ and therefore as tempered distributions. This implies that

$$\begin{aligned} e^{- h^{-2 } t G (h \cdot ) } \Sigma _h \left( \widehat{f }_{\varepsilon } \right) = \widehat{u_h [f_{\varepsilon } ] } \rightarrow \widehat{u_h [f ] } , \end{aligned}$$

where we used Lemma 5.2. Hence $\widehat{u_h [f ] } = e^{- h^{-2 } t G (h \cdot ) } \Sigma _h (\widehat{f } ) = e^{- h^{-2 } t G (h \cdot ) } \widehat{s_h [f ] }$ as tempered distributions, as claimed. We finally prove (51): if we let

$$\begin{aligned} g(\xi , t ; h ) := e^{- t (h^{-2 } G(h \xi ) - |\xi |^2 ) } - 1 , \end{aligned}$$

then g is a $C^{K , \lambda }$ - function of $\xi$ and

$$\begin{aligned} \widehat{u_h [f ] } (\cdot , t ) - \widehat{u } (\cdot , t ) = e^{- t h^{-2 } G(h \cdot ) } \left( \, \widehat{s_h [f ] } - \widehat{f } \, \right) + (g(\cdot , t ; h ) - 1 ) \, e^{ - t |\cdot |^2 } \widehat{f } . \end{aligned}$$

Since $g (\xi , t ; h )$ vanishes to order $|\xi |^{\kappa }$ in $\xi = 0$, by Proposition 5.3(ii), the representation $\widehat{f } = \Lambda _{\widehat{f } }+ \sum _{|\alpha | \le \lceil \kappa \rceil - 1 } c_{\alpha } \delta ^{(\alpha ) }$ from the proof of lemma 3.5 shows that the distribution $g (\cdot , t , h ) \widehat{f }$ can be identified with the locally integrable function $\xi \rightarrow g(\xi , t , h ) \widehat{f }(\xi ) .$ $\square$