Definition 1. d d 1 Cc(R ) := {f : R → C, continuous and with compact support} Ccdef Definition 2. The support of a continuous (!) function is defined as the closure of the set of “relevant points”: supp(f) := {x, f(x) 6= 0}−

Definition 3. (1) translation Tx : Txf(z) = [Txf](z) = f(z − x) (the graph is preserved but moved by the vector x to another position); (2) dilation Dρ (value preserving) and Stρ (mass preserving); (3) involution f 7→ fˇ with fˇ(z) = f(−z); (4) modulation Mω: Multiplication with the character x 7→ exp(2πiωx), i.e. [Mωf](z) := e2πiω·zf(z) (5) Fourier transform F, F −1, to be discussed only later: our normalization will be d that given for f ∈ Cc(R ) by the integral Z Four-intdef (1) F : f 7→ fˆ : fˆ(s) = f(t)e2πis·tdt d R Cbdef Definition 4. d d C ( ) := {f : 7→ , continuous and bounded with the norm kfk = sup d |f(x)| } b R R C ∞ x∈R d d The spaces Cub(R ) and C0(R ) are defined as the subspaces of functions which are uniformly continuous resp. decaying at infinity, i.e. d f ∈ C0(R ) if and only if lim |f(x)| = 0. |x|→∞

d d ∞ d Lemma 1. Characterization of Cub(R ) within Cb(R ) (or even L (R )).

kTxf − fk∞ → 0 for x → 0. d if and only if f ∈ Cub(R ). d We will use the symbol L(C0(R )) for the Banach space of all bounded and linear d operators on the Banach space C0(R ), endowed with the operator norm kT k∞ :=

supkfk∞≤1kT fk∞.

Definition 5. An directed family (a net or sequence) (hα)α∈I in a Banach algebra (B, k · kB) is called a BAI (= bounded approximate identity or “approximate unit” for (B, k · kB)) if lim khα · h − hkB = 0 ∀h ∈ B. α Definition 6. Definition of value preserving dilation operators: d Drho1 (2) Dρf(z) = f(ρ · z), ρ > 0, z ∈ R d
d Definition 7. We denote the dual space of C0(R ), k · k∞ with (M(R ), k · kM ). Some- d times the symbl Mb(R ) is used in order to emphasize that one has “bounded” (regular Borel) measures.

1The capital ”C” stands for continuous, the subscript for compact support

1 2

Definition 8. ∞ ∞ d d X X Md(R ) = {µ ∈ M(R ): µ = ckδtk s.t. |ck| < ∞} k=1 k=1

Definition 9. A sequence Φ = (Tλϕ)λ∈Λ, where ϕ is a compactly supported function d d d (i.e. ϕ ∈ Cc(R )), and Λ = A(Z ) a lattice in R (for some non-singular d × d-matrix) is called a regular BUPU if X ϕ(x − λ) ≡ 1. λ

Definition 10. For any BUPU Φ the Spline-type Quasi-Interpolation operator SpΦ is given by: X f 7→ SpΦ(f) := f(λ)φλ λ∈Λ Definition 11. Definition of dilation operators: d Dρf(z) = f(ρ · z), ρ > 0, z ∈ R d def-BIBOTLIS Definition 12. The Banach space of all “translation invariant linear systems” on C0(R ) is given by 2

d d d d HG(C0(R )) = {T : C0(R ) → C0(R ), bounded, linear : T ◦ Tz = Tz ◦ T, ∀z ∈ R } Definition 13. recall the notion of a FLIP operator: fˇ(z) = f(−z) d ˇ Given µ ∈ M(R ) we define the convolution operator Cµ by: Cµ(f)(z) := µ(Tzf). The reverse mapping R recovers a measure µ = µT from a given translation invariant system T via µ(f) := T (fˇ)(0).

Strhodef Definition 14. The adjoint action of the group R+ on M(Rd) is defined as the family of adjoint operators on M(Rd) via: d def-Stroh (3) Stρµ(f) := µ(Dρf), ∀f ∈ C0(R ), ρ > 0. Definition 15. ALTERNATIVE DESCRIPTION (replaced later >> theorem): Given d −1 µ ∈ M(R ) the uniquely determined measure corresponding to the operator Dρ ◦Cµ ◦Dρ will be denoted by Stρµ.

BanMod Definition 16. A Banach space (B, k · kB) is a Banach module over a Banach algebra (A, k · kA) if one has a bilinear mapping (a, b) 7→ a • b, from A × B into B with

ka • bkB ≤ kakAkbkB ∀ a ∈ A, b ∈ B which behaves like an ordinary multiplication, i.e. is associative, distributive, etc.:

a1 • (a2 • b) = (a1 · a2) • b ∀a1, a2 ∈ A, b ∈ B.

2The letter H in the definition refers to homomorphism [between normed spaces], while the subscript G in the symbol refers to “commuting with the action of the underlying group G = Rd realized by the so-called regular representation, i.e. via ordinary translations 3

dual-alg Definition 17. For any Banach algebra (A, k · kA) the dual space can be turned natu- rally into a A-Banach module via the action d dual-alg (4) [a1 • σ](a) := σ(a1 · a), ∀a, a1 ∈ A, σ ∈ A . Definition 18. A character is a continuous function from a topological group into the torus group T = {z ∈ C | |z| = 1}. In other words, χ is a character if χ(x+y) = χ(x)·χ(y) for all x, y ∈ G. Moreover, since |χ(x)| = 1 one has χ(x) = 1/χ(x) for all x ∈ G. Definition 19. The set of all character is called the dual group, because those characters form an Abelian group under pointwise multiplication (!Exercise!). We write Gˆ for the dual group corresponding to G (the group law written as addition in this case). d Definition 20. The Fourier transform of µ ∈ Mb(R ) is defined by meas-FT-def (5)µ ˆ(s) = µ(χs) 1 d d Li-Def Definition 21. We define L (R ) as the closure of M Cc within (M(R ), k · kM ). meas-support Definition 22. A point x does not belongs to the support of a measure µ ∈ M(Rd) if d there exists some k ∈ Cc(R ) with k(x) = 1, but nevertheless k · µ 6= 0. The complement of this set is denoted by supp(µ). Definition 23. A bounded subset H ⊂ M(Rd) is called (uniformly) tight if for every d ε > 0 there exists k ∈ Cc(R ) such that kµ − k · µkM ≤ ε for al µ ∈ H. d Definition 24. A bounded subset H ⊂ C0(R ) is called (uniformly) tight if for every d ε > 0 there exists h ∈ Cc(R ) such that kh − k · hk∞≤ ε for al h ∈ H. WCdefphi Definition 25. Let ϕ be any non-zero, non-negative function on Rd. 1 d 1 d X WCdefphi1 (6) W (C0, L ) = { f ∈ C0(R ) | ∃(ck)k∈N ∈ ` , (xk)k∈N in R , |f(x)| ≤ ckϕ(x − xk) } k∈N We define 1 d X kf | W (C0, L )(R )k := inf {kck`1 = |ck| } k∈ N WCdefphi1 where the infimum is taken over all “admissible dominations” of f as in (6).

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