Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 687–699

An Integral Operator Representation of

Classical Periodic Pseudodifferential Operators

G. Vainikko

Abstract. In this note we prove that every classical 1-periodic pseudodifferential operator A of order α ∈ ℝ \ ℕ_{0} can be represented in the form

(Au)(t) = ∫_{0}^{1} [κ_{α}^{+}(t − s)a_{+}(t, s) + κ_{α}^{−}(t − s)a_{−}(t, s) + a(t, s)]u(s) ds

where α_{±} and a are C^{∞}-smooth 1-periodic functions and κ_{α}^{±} are 1-periodic functions or distributions with Fourier coefficients κ̂_{α}^{+}(n) = |n|^{α} and κ̂_{α}^{−}(n) = |n|^{α}sign(n) (0 ≠ n ∈ ℤ) with respect to the trigonometric orthonormal basis {e^{in2πt}}_{n∈ℤ} of L^{2}(0, 1). Some explicit formulae for κ_{α}^{±} are given. The case of operators of order α ∈ ℕ_{0} is discussed, too.

Keywords: Classical periodic pseudodifferential operators, periodic integral operators, asymptotic expansions

AMS subject classification: Primary 47 G 30, secondary 47 G 10, 58 G 15

1. Periodic pseudodifferential operators

By H^{λ} (λ ∈ ℝ) we denote the Sobolev space of 1-periodic functions or distributions u having a finite norm

∥u∥_{λ} = ( ∑_{n∈ℤ} \underline{n}^{2λ}|û(n)|^{2})^{\frac{1}{2}}

where

û(n) = ∫_{0}^{1} u(t)e^{−in2πt}dt = ⟨u, e^{−in2πt}⟩

are the Fourier coefficients of u and \underline{n} = max\{1, |n|\}. As usual, L(H^{λ}, H^{μ}) denotes the space of linear bounded operators from H^{λ} into H^{μ}. Every operator A ∈ L(H^{λ}, H^{μ}) is of the form

G. Vainikko: Helsinki Univ. Techn., Inst. Math., P.O. Box 1100, FIN-02015 HUT e-mail: Gennadi.Vainikko@hut.fi

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

u(t) = ∑_{n∈ℤ} û(n)e^{in2πt} ⟼ (Au)(t) = ∑_{n∈ℤ} σ(t, n)û(n)e^{in2πt}  (1)

(and one writes A = Op σ) where

σ(t, n) = e^{−in2πt}Ae^{in2πt}

is called the symbol of A. Indeed, the Fourier series

u(t) = ∑_{n∈ℤ} û(n)e^{in2πt}

of u ∈ H^{λ} converges in H^{λ}, therefore the series

(Au)(t) = ∑_{n∈ℤ} û(n)Ae^{in2πt} = ∑_{n∈ℤ} û(n)σ(t, n)e^{in2πt}

converges in H^{μ}. Clearly, σ(t, n) is 1-periodic in t.

A complex-valued function

σ = σ(t, n) (t ∈ ℝ, n ∈ ℤ)

is called a periodic symbol of degree α (α ∈ ℝ), denoted σ ∈ Σ^{α}, if it is C^{∞}-smooth and 1-periodic in t and satisfies the inequalities

|(\frac{∂}{∂t})^{j}Δ_{n}^{k}σ(t, n)| ≤ c_{jk} \underline{n}^{α−k}  (j, k ∈ ℕ_{0}, t ∈ ℝ, n ∈ ℤ).  (2)

Here ℕ_{0} = {0} ∪ ℕ = {0, 1, 2, …}, and Δ is the (forward) difference operator:

(Δψ)(n) = ψ(n + 1) − ψ(n)

for ψ : ℤ → ℂ. An operator A = Op σ of form (1) with σ ∈ Σ^{α} is called a periodic pseudodifferential operator of order α, denoted A ∈ Op Σ^{α}. This definition originates from [1, 2]. Equivalent definitions can be found in [2 - 4, 12]. It occurs (see, e.g., [12]) that A ∈ L(H^{λ}, H^{λ−α}) for any λ ∈ ℝ if A ∈ Op Σ^{α}.

Introduce a C^{∞}-smooth function h : ℝ → ℝ satisfying

h(k) = \{ \begin{matrix}1 & if k = 0 \\ 0 & if 0 ≠ k ∈ ℤ \end{matrix}

h ∈ S(ℝ), i.e. sup_{ξ∈ℝ} |ξ^{j}h^{(k)}(ξ)| < ∞ (j, k ∈ ℕ_{0})

∀ k ∈ ℕ ∃ h_{k} ∈ S(ℝ) such that h^{(k)}(ξ) = (Δ^{k}h_{k})(ξ) (ξ ∈ ℝ)

(see [12] for a construction of h). The formula

σ(t, ξ) = ∑_{n∈ℤ} σ(t, n)h(ξ − n) (ξ ∈ ℝ)

defines a prolongation σ : ℝ × ℝ → ℂ of σ ∈ Σ^{α}. It occurs that (2) implies the inequalities

|(\frac{∂}{∂t})^{j}(\frac{∂}{∂ξ} )^{k}σ(t, ξ)| ≤ c′_{jk}(1 + |ξ|)^{α−k}  (j, k ∈ ℕ_{0}; t, ξ ∈ ℝ).  (3)

Indeed,

Δ_{ξ}h(ξ − n) = h(ξ + 1 − n) − h(ξ − n) = −\overline{Δ}_{n}h(ξ − n)

where \overline{Δ} is the backward difference operator, thus

(\frac{∂}{∂t})^{j}(\frac{∂}{∂ξ} )^{k}σ(t, ξ) = ∑_{n∈ℤ} (\frac{∂}{∂t})^{j}σ(t, n)h^{(k)}(ξ − n)

= ∑_{n∈ℤ} (\frac{∂}{∂t})^{j}σ(t, n)Δ_{ξ}^{k}h_{k}(ξ − n)

= (−1)^{k} ∑_{n∈ℤ} (\frac{∂}{∂t})^{j}σ(t, n)\overline{Δ}_{n}^{k}h_{k}(ξ − n)

= ∑_{n∈ℤ} h_{k}(ξ − n)(\frac{∂}{∂t})^{j}Δ_{n}^{j}σ(t, n)

(summation by parts on the last step). Since h_{k} ∈ S(ℝ) we have |h_{k}(ξ − n)| ≤ c_{r}(1 + |ξ − n|)^{−r} with any r > 0; we take r > |α − k| + 1. Due to (2), we obtain (3):

|(\frac{∂}{∂t})^{j}(\frac{∂}{∂ξ} )^{k}σ(t, ξ)| ≤ c_{r} c_{jk} ∑_{n∈ℤ}(1 + |ξ − n|)^{−r}\underline{n}^{α−k} ≤ c′_{jk}(1 + |ξ|)^{α−k}.

The converse is also true: if σ : ℝ × ℝ → ℂ satisfies (3), then its restriction to ℝ × ℤ satisfies (2). Thus we may assume that the symbol σ ∈ Σ^{α} is defined and C^{∞}-smooth on ℝ × ℝ, 1-periodic in t and satisfies (3). Nevertheless, only the values on ℝ × ℤ of σ are used to define A = Op σ ∈ Op Σ^{α}.

A symbol σ ∈ Σ^{α} is called classical or polyhomogeneous, denoted σ ∈ Σ_{cl}^{α}, if it admits an asymptotic expansion

σ(t, ξ) ∼ ∑_{j=0}^{∞} σ_{j}(t, ξ),  i.e. σ − ∑_{j=0}^{N−1} σ_{j} ∈ Σ^{α−N} (N ∈ ℕ)  (4)

where σ_{j} ∈ Σ^{α−j} are positively homogeneous of degree α − j in ξ for |ξ| ≥ 1:

σ_{j}(t, τξ) = τ^{α−j}σ_{j}(t, ξ) (|ξ| ≥ 1, τ ≥ 1).

Clearly,

σ_{j}(t, ξ) = \{ \begin{matrix}σ_{j}(t, 1)ξ^{α−j} =: a_{j}^{+}(t)ξ^{α−j} & for ξ ≥ 1 \\ σ_{j}(t, −1)|ξ|^{α−j} =: a_{j}^{−}(t)|ξ|^{α−j} & for ξ ≤ −1 \end{matrix}  (5)

with a_{j}^{±} ∈ C_{1}^{∞}(ℝ) where C_{1}^{∞}(ℝ) denotes the set of 1-periodic C^{∞}-smooth functions on ℝ. The corresponding A = Op σ is called a classical periodic pseudodifferential operator of order α, denoted A ∈ Op Σ_{cl}^{α}. It follows from (4) and (5) that

A ∼ ∑_{j=0}^{∞} A_{j}, i.e. A − ∑_{j=0}^{N−1} A_{j} ∈ Op Σ^{α−N} (N ∈ ℕ)

where

\begin{aligned}A_{j} & = [a_{j}^{+}(t)P^{+} + a_{j}^{−}(t)P^{−}]L^{α−j} ∼ Op σ_{j} \\ P^{+}u & = ∑_{n≥0} û(n)e^{in2πt} \\ P^{−}u & = ∑_{n<0} û(n)e^{in2πt} \\ L^{λ}u & = ∑_{n∈ℤ} \underline{n}^{λ}û(n)e^{in2πt} (λ ∈ ℝ). \end{aligned}

Let us comment on the polyhomogenuity of a symbol. It occurs that a symbol σ ∈ Σ^{α} belongs to Σ_{cl}^{α} if and only if |ξ|^{−α}σ(t, ξ) behaves in a regular manner as ξ → ±∞, or equivalently, σ_{∗}(t, η) = |η|^{α}σ(t, \frac{1}{η}) with η = \frac{1}{ξ} behaves in a regular manner as η → ±0. Namely, if σ_{∗} has C^{∞}-smooth continuations to η = +0 and η = −0, then the Taylor expansions

\begin{aligned}σ_{∗}(t, η) & = ∑_{j=0}^{N−1} a_{j}^{+}(t)η^{j} + O(η^{N})  (η → +0, a_{j}^{+}(t) = \frac{1}{j!}(\frac{∂}{∂η} )^{j}σ_{∗}(t, η)|_{η=+0}) \\ σ_{∗}(t, η) & = ∑_{j=0}^{N−1} a_{j}^{−}(t)η^{j} + O(η^{N})  (η → −0, a_{j}^{−}(t) = \frac{1}{j!}(\frac{∂}{∂η} )^{j}σ_{∗}(t, η)|_{η=−0}) \end{aligned}

hold true for all N ∈ ℕ. Returning to ξ = \frac{1}{η} and σ(t, ξ) = |ξ|^{α}σ_{∗}(t, ξ^{−1}) we have

\begin{aligned}σ(t, ξ) & = ∑_{j=0}^{N−1} a_{j}^{+}(t)|ξ|^{α−j} + O(ξ^{α−N})  (ξ → +∞) \\ σ(t, ξ) & = ∑_{j=0}^{N−1} a_{j}^{−}(t)|ξ|^{α−j} + O(ξ^{α−N})  (ξ → −∞) \end{aligned}

and it can be checked that by those a_{±} the asymptotic expansion (4) - (5) is defined.

2. Integral operator representation of periodic pseudodifferential operators

Here we follow some ideas from [3, 5, 8 - 10]. For A ∈ Op Σ^{α} (α < −1) and u ∈ H^{0} = L^{2}(0, 1) we have

(Au)(t) = ∑_{n∈ℤ} σ(t, n)û(n)e^{in2πt}

= ∑_{n∈ℤ} σ(t, n)e^{in2πt} ∫_{0}^{1} u(s)e^{−in2πs}ds

= ∫_{0}^{1} ∑_{n∈ℤ} σ(t, n)e^{in2π(t−s)}u(s) ds

= ∫_{0}^{1} K(t, t − s)u(s) ds

where the series

K(t, s) = ∑_{n∈ℤ} σ(t, n)e^{in2πs}

converges uniformly in t, s ∈ ℝ due to the estimate |σ(t, n)| ≤ c_{00}\underline{n}^{α} (see (2)). Thus K(t, s) is continuous on ℝ × ℝ. Moreover, K(t, s) is C^{∞}-smooth for s ∈ ℝ \ ℤ (and then K(t, t − s) is C^{∞}-smooth for t − s ∉ ℤ). Indeed, consider the product

(e^{−i2πs} − 1)K(t, s) = ∑_{n∈ℤ} σ(t, n)(e^{i(n−1)2πs} − e^{in2πs})

= ∑_{n∈ℤ} [σ(t, n + 1) − σ(t, n)]e^{in2πs}

= ∑_{n∈ℤ} [Δσ(t, n)]e^{in2πs}.

Repeating the multiplications by (e^{−i2πs} − 1) we obtain

(e^{−i2πs} − 1)^{l}K(t, s) = ∑_{n∈ℤ} [Δ^{l}σ(t, n)]e^{in2πs} (l ∈ ℕ).

Now estimate (2) yields that (e^{−i2πs} −1)^{l}K(t, s) is l-times continuously differentiable on ℝ×ℝ. Since l is arbitrary, K(t, s) is infinitely smooth for (t, s) satisfying e^{−i2πs} −1 ≠ 0, i.e. for s ∈ ℝ\ℤ. Also the case α ∈ [−1, 0) can be treated, but then K(t, t−s) is weakly singular for t = s.

For u ∈ H^{l} (l ∈ ℕ_{0}) integration by parts yields

û(n) = ∫_{0}^{1} u(s)e^{−in2πs}ds = \frac{1}{(2πin)^{l}} ∫_{0}^{1} u^{(l)}(s)e^{−in2πs}ds

and

(Au)(t) = ∫_{0}^{1} K_{l}(t, t − s)u^{(l)}(s) ds, K_{l}(t, s) = ∑_{N∈ℤ} \frac{σ(t, n)}{(2πin)^{l}} e^{in2πs}.

Now already for σ ∈ Σ^{α} with α < l − 1, the series converges uniformly and defines a continuous kernel K_{l} on ℝ × ℝ; for s ∈ ℝ \ ℤ, K_{l}(t, s) is again C^{∞}-smooth.

One can try to represent K and K_{l} in the form of products

K(t, t − s) = a(t, s)κ(t − s)  and  K_{l}(t, t − s) = a_{l}(t, s)κ_{l}(t − s),

respectively, where a and a_{l} are C^{∞}-smooth on the whole ℝ × ℝ whereas κ and κ_{l} are C^{∞}-smooth on ℝ \ ℤ. With some specifications we shall succeed in the case of classical periodic pseudodifferential operators. For a general (non-classical) periodic pseudodifferential operator a similar representations does not exist.

We point out also the following inverse result from [11].

Theorem 1. An integral operator defined by

(Au)(t) = ∫_{0}^{1} κ(t − s)a(t, s)u(s) ds

with a C^{∞}-smooth 1-biperiodic function a and 1-periodic function or distribution κ belongs to Op Σ^{α} if κ satisfies

|Δ^{k}κ̂(n)| ≤ c_{k}\underline{n}^{α−k} (k ∈ ℕ_{0}, n ∈ ℤ)

or, equivalently, if the extended function κ̂ : ℝ → ℂ (defined by κ̂(ξ) = ∑_{n∈ℤ} κ̂(n)h(ξ − n) or in some other way) satisfies

|(\frac{d}{dξ} )^{k}κ̂(ξ)| ≤ c′_{k}(1 + |ξ|)^{α−k} (k ∈ ℕ_{0}, ξ ∈ ℝ).

Thereby A has asymtotic expansions A ∼ ∑_{j=0}^{∞} A_{j} with

(A_{j}u)(t) = a_{j}(t) ∫_{0}^{1} κ_{j}(t − s)u(s) ds = a_{j}(t) ∑_{n∈ℤ} κ̂_{j}(n)û(n)e^{in2πt}

where

\begin{matrix}κ̂_{j}(n) & = \frac{1}{j!}Δ^{j}κ̂(n) (n ∈ ℤ) \\ a_{j}(t) & = (\frac{1}{2πi} \frac{∂}{∂s} − (j − 1))(\frac{1}{2πi} \frac{∂}{∂s} − (j − 2)) … (\frac{1}{2πi} \frac{∂}{∂s} − 1)\frac{1}{2πi} \frac{∂}{∂s}a(t, s)|_{s=t} \end{matrix} \}

respectively

\begin{matrix}κ̂_{j}(n) & = \frac{1}{j!}(\frac{d}{dξ} )^{j}κ̂(ξ)|_{ξ=n} (n ∈ ℤ) \\ a_{j}(t) & = (\frac{1}{2πi} \frac{∂}{∂s})^{j}a(t, s)|_{s=t}. \end{matrix} \}

3. Integral operator representation of classical periodic pseudodifferential operators

Here we first formulate and at the end prove the main results of the paper.

Theorem 2. Every operator A ∈ Op Σ_{cl}^{α} with α ∈ ℝ \ ℕ_{0} can be represented in the form

(Au)(t) = ∫_{0}^{1} [κ_{α}^{+}(t − s)a_{+}(t, s) + κ_{α}^{−}(t − s)a_{−}(t, s) + a(t, s)]u(s) ds  (6)

where a_{±}, a ∈ C_{1}^{∞}(ℝ × ℝ), i.e. a_{±}, a are C^{∞}-smooth and 1-periodic with respect to both arguments and κ_{α}^{±} are 1-periodic functions or distributions defined by their Fourier coefficients

\begin{matrix}κ̂_{α}^{+}(n) & = |n|^{α} \\ κ̂_{α}^{−}(n) & = |n|^{α}sign(n) \end{matrix} \}  (0 ≠ n ∈ ℤ).  (7)

Conversely, every integral operator of form (6) − (7) with a_{±}, a ∈ C_{1}^{∞}(ℝ × ℝ) belongs to Op Σ_{cl}^{α}.

Note that (7) define κ_{α}^{±} uniquely up to a constant addend κ̂_{α}^{±}(0). Changing κ̂_{α}^{±}(0), only the coefficient a ∈ C_{1}^{∞}(ℝ × ℝ) changes in (6). In (6), the integral means the usual Lebesgue integral for α < 0 and u ∈ H^{0} = L^{2}(0, 1). For α ≥ 0 and u ∈ H^{μ} (μ > α + \frac{1}{2}) the integral can be understood as the dual product between H^{μ} and H^{−μ}, since κ_{α}^{±} ∈ H^{−μ}. The case of u ∈ H^{λ} with an arbitrary λ ∈ ℝ can be understood through the approximation of u by smooth functions, e.g.

Au = lim_{N→∞} AP_{N}u, P_{N}u = ∑_{|n|≤N} û(n)e^{in2πt}.

Here P_{N}u → u in H^{λ} and AP_{N}u → Au in H^{λ−α} for u ∈ H^{λ} (recall that A ∈ L(H^{λ}, H^{λ−α}) for any λ ∈ ℝ if A ∈ Op Σ^{α}).

Now consider the case α ∈ ℕ_{0} excluded from Theorem 2.

Theorem 3. Every operator A ∈ Op Σ_{cl}^{α} with α = m ∈ ℕ_{0} has the representation

\begin{aligned}(Au)(t) & = ∑_{j=0}^{m} [c_{j}^{+}(t)u^{(m−j)}(t) + c_{j}^{−}(t)(H_{0}u^{(m−j)})(t)] \\ & + ∫_{0}^{1} [κ_{−1}^{+}(t − s)a_{+}(t, s) + κ_{−1}^{−}(t − s)a_{−}(t, s) + a(t, s)]u(s) ds \end{aligned}  (8)

where c_{j}^{±} ∈ C_{1}^{∞}(ℝ) and a_{±}, a ∈ C_{1}^{∞}(ℝ × ℝ), H_{0} is the Hilbert transformation

(H_{0}u)(t) = \frac{1}{i} p.v. ∫_{0}^{1} cot π(s − t)u(s) ds = ∑_{n≥1} û(n)e^{in2πt} − ∑_{n≤−1} û(n)e^{in2πt},

and

κ_{−1}^{+}(t) = −2 log | sin πt|  (9)
κ_{−1}^{−} is the 1-periodic extension of t ↦ −2πit from [0, 1) to ℝ  (10)

(these functions satisfy (7) with α = −1).

Remark 1. Using the periodic Dirac delta function and its derivatives one can also (8) represent as an integral operator.

Remark 2. Clearly, Σ_{cl}^{α} ⊂ Σ_{cl}^{α+1}, therefore we actually have different possible integral operator representations of an operator A ∈ Op Σ_{cl}^{α}. For instance, A ∈ Op Σ_{cl}^{−m} with an m ∈ ℕ can be represented in the form (6) with α = −1 and κ_{−1}^{±} defined in (9)

- (10); the order −m of the operator can be discovered by properties of the coefficients a_{±}:

(\frac{∂}{∂s})^{j}a_{±}(t, s)|_{s=t} = 0 (t ∈ ℝ; j = 0, …, m − 2).

Operators of type (6) have been examined in [6, 12]. They often appear solving boundary integral equations on closed curves (see, e.g., [5 - 7, 12]).

Proof of Theorem 2. Let A ∈ Op Σ_{cl}^{α}, i.e. its symbol σ(t, ξ) has the asymptotic expansion (4) with σ_{j} of form (5). We regularize the functions |ξ|^{β} in the neighbourhood of ξ = 0 putting

\begin{matrix}ϕ_{β}(0) & = 0 \\ ϕ_{β}(ξ) & = ϕ_{0}(ξ)|ξ|^{β} (ξ ∈ ℝ \ \{0\}) \end{matrix} \}

where ϕ_{0} ∈ C^{∞}(ℝ) satisfies

ϕ_{0}(ξ) = \{ \begin{matrix}1 & for |ξ| ≥ 1 \\ 0 & for |ξ| ≤ \frac{1}{2}. \end{matrix}

Thus we have

σ(t, ξ) ∼ ∑_{j=0}^{∞} [b_{j}^{+}(t)ϕ_{α−j}(ξ) + b_{j}^{−}(t)ϕ_{α−j}(ξ)sign(ξ)]  (11)

where

b_{j}^{+}(t) = \frac{1}{2}[a_{j}^{+}(t) + a_{j}^{−}(t)]  and  b_{j}^{−}(t) = \frac{1}{2}[a_{j}^{+}(t) − a_{j}^{−}(t)].

On the other hand, by Theorem 1 the integral operator defined in (6) is a (classical) periodic pseudodifferential operator with the symbol σ̃ having the asymptotical expansion

\begin{aligned}σ̃(t, ξ) & ∼ ∑_{j=0}^{∞} \frac{1}{j!}(\frac{∂}{∂ξ} )^{j}ϕ_{α}(ξ)(\frac{1}{2πi} \frac{∂}{∂s})^{j}a_{+}(t, s)|_{s=t} \\ & + ∑_{j=0}^{∞} \frac{1}{j!}(\frac{∂}{∂ξ} )^{j}ϕ_{α}(ξ)sign(ξ)(\frac{1}{2πi} \frac{∂}{∂s})^{j}a_{−}(t, s)|_{s=t}. \end{aligned}

Representation (6) of A ∈ Op Σ_{cl}^{α} takes place if σ ∼ σ̃, i.e. σ −σ̃ ∈ Σ^{−∞}. For α ∈ ℝ\ℕ_{0} this means that a_{±} ∈ C_{1}^{∞}(ℝ × ℝ) satisfy

\frac{α(α − 1) ⋯ (α − j + 1)}{j!}(\frac{1}{2πi} \frac{∂}{∂s})^{j}a_{±}(t, s)|_{s=t} = b_{j}^{±}(t) (t ∈ ℝ, j ∈ ℕ_{0}).

Thus, to prove Theorem 2, we simply have to solve the following elementary problem: given b_{j} ∈ C_{1}^{∞}(ℝ) (j ∈ ℕ_{0}), construct a ∈ C_{1}^{∞}(ℝ × ℝ) such that

(\frac{∂}{∂s})^{j}a(t, s)|_{s=t} = b_{j}(t) (t ∈ ℝ, j ∈ ℕ_{0}).

A solution may be given by a regularization and periodization of the Taylor series:

a(t, s) = ∑_{l=0}^{∞} \frac{b_{l}(t)}{l!}[χ(s − t)]^{l}ψ_{N_{l}}(s − t).

Here χ ∈ C_{1}^{∞}(ℝ) satisfies χ(s) = s for |s| ≤ \frac{1}{4}, and ψ_{N} ∈ C_{1}^{∞}(ℝ) (N ∈ ℕ) satisfies

ψ_{N}(t) = \{ \begin{matrix}1 & for |t| ≤ \frac{1}{8N} \\ 0 & for \frac{1}{4N} ≤ |t| ≤ \frac{1}{2}. \end{matrix}

More concretely, we define

ψ_{N}(t) = ∑_{j∈ℤ} ψ(Nt + j) where ψ ∈ C^{∞}(ℝ) with ψ(t) = \{ \begin{matrix}1 & for |t| ≤ \frac{1}{8} \\ 0 & for |t| ≥ \frac{1}{4}. \end{matrix}

The numbers N_{l} ≥ 1 should be chosen so that the series itself and the series after applying (\frac{∂}{∂t})^{j}(\frac{∂}{∂s})^{k} (j, k ∈ ℕ_{0}) will converge uniformly for t, s ∈ ℝ. A sufficient condition is given by N_{l} ≥ d_{l} where

d_{l} = max_{0≤n≤l} max_{0≤t≤1} |b_{l}^{(n)}(t)|.

Indeed, applying (\frac{∂}{∂t})^{j}(\frac{∂}{∂s})^{k} we obtain a finite number of series of the type

∑_{l=p}^{∞} \frac{b_{l}^{(n)}(t)}{(l − p)!}[χ(s − t)]^{l−p}ψ_{N_{l}}^{(q)}(s − t) (n ≤ j; p, q ≤ j + k)

(notice that χ(s) = s for s ∈ supp ψ_{N_{l}} ∩ [−\frac{1}{2}, \frac{1}{2}]). For l ≥ n the members of the last series can be estimated by

\frac{d_{l}}{(l − p)!} (4N_{l})^{−(l−p)}c_{q}N_{l}^{q} ≤ \frac{c_{q}}{(l − p)!} 4^{−(l−p)}N_{l}^{−l+p+q+1}

guaranteeing uniform convergence of the series Proof of Theorem 3. For α = m ∈ ℕ_{0}, we present (11) as the sum

σ = σ_{m} + σ^{[m]}

with

σ_{m}(t, ξ) = ∑_{j=0}^{m} [b_{j}^{+}(t)ϕ_{m−j}(ξ) + b_{j}^{−}(t)ϕ_{m−j}(ξ)sign(ξ)]

and

σ^{[m]}(t, ξ) = ∑_{j=m+1}^{∞} [b_{j}^{+}(t)ϕ_{m−j}(ξ) + b_{j}^{−}(t)ϕ_{m−j}(ξ)sign(ξ)]

= ∑_{j=0}^{∞} [b_{j+m+1}^{+}(t)ϕ_{−1−j}(ξ) + b_{j+m+1}^{−}(t)ϕ_{−1−j}(ξ)sign(ξ)]

and the representation (8) - (10) for A ∈ Op Σ_{cl}^{m} follows immediately from Theorem 2. Thereby,

c_{j}^{±}(t) = (2πi)^{j−m}b_{j}^{±}(t) (0 ≤ j ≤ m)

and the theorem is proved

Remark 3. As it can be seen from the proof, also some non-classical periodic pseudodifferential operators have an integral operator representation similar to (6). Namely, if σ ∈ Σ^{α} has an asymptotic expansion

σ(t, ξ) ∼ ∑_{j=0}^{∞} [a_{j}^{+}(t)γ^{(j)}(ξ) + a_{j}^{−}(t)γ^{(j)}(ξ)sign(ξ)]

where

\begin{matrix}a_{j}^{±} & ∈ C_{1}^{∞}(ℝ) \\ γ & ∈ C^{∞}(ℝ) with |γ^{(j)}(ξ)| ≤ c_{j}(1 + |ξ|)^{α−j} (ξ ∈ ℝ, j ∈ ℕ_{0}) \end{matrix} \} ,

then A = Op σ can be represented in the form

(Au)(t) = ∫_{0}^{1} [κ_{+}(t − s)a_{+}(t, s) + κ_{−}(t − s)a_{−}(t, s) + a(t, s)]u(s) ds

where

a_{±}, a ∈ C_{1}^{∞}(ℝ × ℝ)  and  \{ \begin{matrix}κ̂_{+}(n) & = γ(n) \\ κ̂_{−}(n) & = γ(n)sign(n) \end{matrix} (0 ≠ n ∈ ℤ).

4. Functions κ_{α}^{±}

Here we present some formulae of functions κ_{α}^{±} satisfying (7). For α = −1 these formulae are well-known (see (9) - (10)). Consider the case −1 < α < 0. Introduce the function

κ_{α}(t) = t^{|α|−1} + ∑_{j=1}^{∞}[(t + j)^{|α|−1} − γ_{j}]  (0 < t ≤ 1, −1 < α < 0)  (12)

where

γ_{j} = ∫_{0}^{1}(t + j)^{|α|−1}dt = \frac{1}{|α|}[(j + 1)^{|α|} − j^{|α|}].

Note that the series in (12) converges uniformly in t ∈ [0, 1], since γ_{j} as the mean value of (t + j)^{|α|−1} in [0, 1] has a representation γ_{j} = (t_{j} + j)^{|α|−1} with a t_{j} ∈ (0, 1), and

\begin{matrix}(t + j)^{|α|−1} − γ_{j} & = (t + j)^{|α|−1} − (t_{j} + j)^{|α|−1} = (|α| − 1)(t′_{j} + j)^{|α|−2}(t − t_{j}) \\ |(t + j)^{|α|−1} − γ_{j}| & ≤ (1 − |α|)j^{|α|−2} \end{matrix} \}

where t′_{j} ∈ (t, t_{j}) ⊂ (0, 1). Clearly, also the series obtained after differentiations converge uniformly. Thus, κ_{α} ∈ C^{∞}(0, 1]. Moreover, κ_{α} is decreasing and 0 < κ_{α}(t) < t^{|α|−1} (0 < t ≤ 1).

Define

κ_{α}^{±}(t) = γ_{α}^{±}[κ_{α}(t) ± κ_{α}(1 − t)] (0 < t < 1) (13)

where

γ_{α}^{±} = \frac{1}{c_{α}^{±}}
c_{α}^{+} = 2(2π)^{α}Γ(|α|) cos \frac{|α|π}{2}
c_{α}^{−} = −2i(2π)^{α}Γ(|α|) sin \frac{|α|π}{2}

and

Γ(β) = ∫_{0}^{∞}t^{β−1}e^{−t}dt (0 < β < 1)

is the Euler function. We preserve the designations κ_{α}^{±} also for the 1-periodic extensions of those functions and assert that

\begin{matrix}κ̂_{α}^{+}(n) & = |n|^{α} \\ κ̂_{α}^{−}(n) & = |n|^{α}sign(n) \end{matrix} \}  (0 ≠ n ∈ ℤ, −1 < α < 0).  (14)

To prove this we first find the Fourier coefficients of κ_{α}(t) (0 < t ≤ 1) which we also regard as extended up to a 1-periodic function. Clearly,

\begin{matrix}κ̂_{α}(0) & = ∫_{0}^{1} t^{|α|−1}dt = \frac{1}{|α|} \\ κ̂_{α}(n) & = ∑_{j=0}^{∞} ∫_{0}^{1}(t + j)^{|α|−1}e^{−in2πt}dt (0 ≠ n ∈ ℤ) \end{matrix} \} .

With the changes of variables

\begin{matrix}t + j & = s \\ 2π|n|s & = τ \end{matrix} \}

we have

κ̂_{α}(n) = ∑_{j=0}^{∞} ∫_{j}^{j+1}s^{|α|−1}e^{−in2πs}ds

= ∫_{0}^{∞}s^{|α|−1}e^{−in2πs}ds

= (2π|n|)^{α} ∫_{0}^{∞}τ^{|α|−1}e^{−sign(n)iτ} dτ.

It is known that

∫_{0}^{∞}τ^{β−1}e^{±iτ} dτ = e^{±i \frac{π}{2} β}Γ(β) (0 < β < 1)

(see, e.g., [13: p. 69] for a proof). Thus

κ̂_{α}(n) = (2π)^{α}Γ(|α|)e^{−isign(n) \frac{π}{2} |α|}|n|^{α} (0 ≠ n ∈ ℤ).

The Fourier coefficient of functions v and w such that w(t) = v(1 − t) are related by ŵ(n) = v̂(−n). Therefore, the Fourier coefficients of κ_{α}^{±} defined by (13) are as follows:

κ̂_{α}^{±}(n) = γ_{α}^{±}(2π)^{α}Γ(|α|)(e^{−isign(n) \frac{π}{2} |α|} ± e^{isign(n) \frac{π}{2} |α|})|n|^{α} (0 ≠ n ∈ ℤ).

This results to (14).

Now we have formulae of κ_{α}^{±} satisfying (7) for −1 ≤ α < 0. The following obvious remark makes possible to extend the result for other α ∈ ℝ.

Remark 4. The formulae

\begin{matrix}κ_{α−1}^{+}(t) & = 2πi ∫_{0}^{t}[κ_{α}^{−}(s) − κ̂_{α}^{−}(0)] ds \\ κ_{α−1}^{−}(t) & = 2πi ∫_{0}^{t}[κ_{α}^{+}(s) − κ̂_{α}^{+}(0)] ds \end{matrix} \} (α < 0)

and

\begin{matrix}κ_{α+1}^{+}(t) & = \frac{1}{2πi} \frac{d}{dt}κ_{α}^{−}(t) \\ κ_{α+1}^{−}(t) & = \frac{1}{2πi} \frac{d}{dt}κ_{α}^{+}(t) \end{matrix} \} (α ∈ ℝ)

hold where \frac{d}{dt} means the periodic distribution derivative.

Acknowledgement The author thanks Professor M. S. Agranovich for a useful discussion.

An Integral Operator Representation 699

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Received 22.09.1998