Inhomogeneous poly-scale refinement type equations and Markov operators with perturbations

AbstractGiven measure spaces $${(\Omega_{1}, \mathcal{A}_{1}, \mu_{1}),...,(\Omega_{N}, \mathcal{A}_{N}, \mu_{N}),}$$(Ω1,A1,μ1),...,(ΩN,AN,μN), functions $${\varphi_{1}: \mathbb{R}^{m} \times \Omega_{1} \rightarrow \mathbb{R}^{m},...,\varphi_{N}: \mathbb{R}^{m} \times \Omega_{N} \rightarrow \mathbb{R}^{m}}$$φ1:Rm×Ω1→Rm,...,φN:Rm×ΩN→Rm and $${g: \mathbb{R}^{m} \rightarrow \mathbb{R}}$$g:Rm→R, we present results on the existence of solutions $${f: \mathbb{R}^{m} \rightarrow \mathbb{R}}$$f:Rm→R of the inhomogeneous poly-scale refinement type equation of the form
$$f(x) = \sum_{n=1}^{N} \int_{\Omega_{n}}|{\rm det}(\varphi_{n})^{\prime} _{x}(x, \omega_{n})|f(\varphi_{n}(x, \omega_{n}))d\mu_{n}(\omega_{n}+g(x)$$f(x)=∑n=1N∫Ωn|det(φn)x′(x,ωn)|f(φn(x,ωn))dμn(ωn+g(x)in some special classes of functions. The results are obtained by Banach fixed point theorem applied to a perturbed Markov operator connected with the considered inhomogeneous poly-scale refinement type equation.

in some special classes of functions. The results are obtained by Banach fixed point theorem applied to a perturbed Markov operator connected with the considered inhomogeneous poly-scale refinement type equation.

Introduction
Fix m, N ∈ N, measure spaces (Ω 1 , A 1 , µ 1 ), . . . , (Ω N , A N , µ N ) and functions φ 1 : R m ×Ω 1 → R m , . . . , φ N : R m ×Ω N → R m , g : R m → R. We are interested in solutions f : R m → R of the inhomogeneous poly-scale refinement type equation dµ n (ω n ) + g(x) (1.1) then equation (1.1) becomes the inhomogeneous poly-scale refinement type equation Taking g = 0 in (1.3) we obtain the following homogeneous poly-scale refinement type equation This equation extends the discrete poly-scale refinement equation which has been studied in [5,26,27]. If N = 1, then equation (1.3) reduces to the inhomogeneous refinement type equation . (1.4) Inhomogeneous refinement equations are motivated by constructions of multiwavelets, multichannel filters and constructions of wavelets on a finite interval (see, e.g., [4,10,23]). Various inhomogeneous forms of equation (1.4) have been investigated in [6,12,24,25]. Several problems from different areas of pure and applied mathematics lead to the problem of the existence of nontrivial Lebesgue integrable solutions of refinement type equation (1.4) with g = 0, i.e., the homogeneous refinement type equation (for more details see the survey [15] and the references therein). It turns out that in some applications, continuous and bounded or continuous and compactly supported solutions of homogeneous refinement equations are important. Such solutions have significant applications in wavelet theory, approximation theory, theory of subdivision schemes, computer graphics, physics, combinatorial number theory and many others (see, e.g., [2,7,8,9,18,19,22]). From the point of view of applications, all results on the existence of "good" solutions of homogeneous as well as of inhomogeneous refinement equations are very important (see [3] where it is showed how nonexistence of "good" solutions of a refinement equation can lead to anomalous behavior of numerical methods for a construction of wavelets). Let us note that refinement equations always have plenty of "bad" solutions, even extremely strange (see, e.g., [1,15,20]).
Vol. 17 (2015) Inhomogeneous refinement type equations 509 Inhomogeneous refinement type equations 3 To find conditions under which homogeneous refinement type equation (1.5) has a nontrivial Lebesgue integrable solution is rather difficult; some results in this direction can be found in [14,21]. To the best of our knowledge, there is no result concerning the existence of a nontrivial Lebesgue integrable solution of the homogeneous poly-scale refinement type equation Nevertheless, it is possible to formulate conditions under which homogeneous poly-scale refinement type equation (1.6) has no nontrivial Lebesgue integrable solution (see [13] for the case where N = 1). Therefore, if homogeneous poly-scale refinement type equation (1.6) has no nontrivial Lebesgue integrable solution, we can ask if its inhomogeneous counterpart (1.1), obtained by adding to the right-hand side of (1.6) a perturbation function g, has such a solution.
If F is a given class of functions, then the existence of a solution f ∈ F of inhomogeneous poly-scale refinement type equation (1.1) is a consequence of the existence of a fixed point of the operator P : F → F given by It turns out that the operator P happens (under suitable assumptions) to be the Markov operator in the case where g = 0; for the definition of the Markov operator and more information on it see [17]. Asymptotically stability of Markov operators has been explored in [21] to study the problem of the existence of nontrivial Lebesgue integrable solutions of homogeneous refinement type equation (1.5). In this paper, we are going to examine the Banach fixed point theorem to obtain results on the existence of a pth power Lebesgue integrable solution as well as a continuous and bounded solution of equation (1.1) and of its special case (1.3). Thus we are looking for conditions, on the spaces (Ω 1 , A 1 , µ 1 ), . . . , (Ω N , A N , µ N ) and the given functions φ 1 , . . . , φ N and g, guaranteeing that the operator P is well defined and satisfies assumptions of the Banach fixed point theorem.

Notation
Given a real number p ≥ 1, we write L p to denote the Banach space of all pth power Lebesgue measurable functions f : R m → R with the standard norm here and throughout, l m denotes the Lebesgue measure on R m . Given f ∈ L p , it is easy to see that the set it is called the support of f and if it is compact we say that f is compactly supported. We write C B to denote the Banach space of all continuous and bounded functions f : R m → R with the supremum norm Since R m is a noncompact space, we can consider the subspace C C ⊂ C B of all functions f : R m → R with compact support; in contrast to the definition of the support of pth power Lebesgue measurable functions, we say that a continuous function f : R → R has a compact support if the set called the support of f , is compact. The space C C endowed in the supremum norm is not, in general, complete. The completion of C C is the space C 0 consisting of all those continuous functions f : The symbol B will denote the family of all Borel subsets of R m . From now on we assume that (Ω 1 , A 1 , µ 1 ), . . . , (Ω N , A N , µ N ) are complete finite measure spaces, and P denotes the operator given by (1.7).

L 1 -solutions
Throughout this section we assume that g ∈ L 1 .
As it was mentioned earlier, our aim is to use the Banach fixed point theorem to the operator P defined by (1.7). For this purpose we need to know that P transforms the space L 1 into itself. Before we give conditions on φ n 's under which P ( Vol. 17 (2015) Inhomogeneous refinement type equations 511 Inhomogeneous refinement type equations 5

is a Lebesgue integrable function, then so is the function
Conditions (3.1) and (3.2) imply that both functions φ n and det(φ n ) ′ x are B ⊗ A n -measurable (see [11] or [16]).
Since A n is complete, we conclude from the Fubini theorem that the function is µ n -integrable for almost all x ∈ R m and that the function ∫ is Lebesgue integrable. Fix now an arbitrary Lebesgue integrable function f : R m → R. Since both the functions f + and f − are nonnegative and Lebesgue integrable, we conclude that both the functions ∫ are Lebesgue integrable. In consequence, the function ∫ is also Lebesgue integrable. Finally, the function given by (3.4) is Lebesgue integrable, because it is a finite sum of Lebesgue integrable functions.
Then for every n ∈ {1, . . . , N} there exists a Lebesgue measurable set C n such that l m (C n ) = 0 and µ n (φ −1 n (B) x ) = 0 for every x ̸ ∈ C n . Hence, if i.e.,f satisfies (1.1) for almost all x ∈ R m .
(iii) Fix f ∈ L 1 and choose two representatives f 1 and f 2 of f . By assertion (i) we infer that both the functions P f 1 and P f 2 are Lebesgue integrable. Now by the same arguments as in the proof of assertion (ii) we conclude that P f 1 (x) = P f 2 (x) for almost all x ∈ R m .
Observe that assertion (ii) of the above lemma says that the definition of L 1 -solutions of equation (1.1) is well posed.
In the case where the functions φ n 's are of the form (1.2) we do not need any special condition guaranteeing that P : L 1 → L 1 . More precisely, we have the following observation.  Inhomogeneous refinement type equations 513 Inhomogeneous refinement type equations 7 satisfies conditions (3.1)-(3.3). If ∑ N n=1 µ(Ω n ) < 1, then equation (1.1) has exactly one L 1 -solution f and for every f 0 ∈ L 1 the sequence (P n f 0 ) n∈N converges to f in L 1 . Moreover, and if supp g ⊂ Z, then supp f ⊂ Z; in particular, if Z is compact, then f is compactly supported.
Proof. By Lemma 3.1 we have P : The Banach fixed point theorem completes the proof of the main part of the theorem.
(i) Clearly, both the sets Taking the Fourier transform of both sides of equation (1.1), we obtain for every y ∈ R. Hence for y = 0 we get [ is a closed subspace of L 1 . Fix h ∈ X. By (3.5) we have supp(h • φ n ) ⊂ Z for every n ∈ {1, . . . , N}. This jointly with supp g ⊂ Z implies supp P h ⊂ Z and, in consequence, we obtain P : X → X.  3) has exactly one L 1 -solution f and for every f 0 ∈ L 1 the sequence (P n f 0 ) n∈N converges to f in L 1 . Moreover, for all ω n ∈ Ω n and n ∈ {1, . . . , N} (3.6) and if supp g ⊂ Z, then supp f ⊂ Z; in particular, if Z is compact, then f is compactly supported.
In the case where g = 0, Theorem 3.3 says that the trivial function is the only L 1 -solution of equation (1.6). To see it in another way suppose that, on the contrary, f is a nontrivial L 1 -solution of equation (1.6). Then which is impossible.

L p -solutions
Throughout this section we assume that g ∈ L p with p > 1.
Recall that f ∈ L p is called an L p -solution of equation (1.1), if every representative of f satisfies (1.1) for almost all x ∈ R m with respect to l m .
We begin with a counterpart of Lemma 3.1 for the space L p . Then Proof. (i) Fix n ∈ {1, . . . , N} and pth power Lebesgue integrable function f : R m → R. In the same way as in the proof of assertion (i) of Lemma 3.1, is L m ⊗ A n -measurable. By the Minkowski inequality for integrals and (4.1) which shows that the function ∫ is pth power Lebesgue integrable. In a similar way we can prove that the function ∫ is pth power Lebesgue integrable. Consequently, the function ∫ is pth power Lebesgue integrable. Finally, the function given by (3.4) is pth power Lebesgue integrable, because it is a finite sum of pth power Lebesgue integrable functions. Assertions (ii) and (iii) can be proved in the same manner as assertions (ii) and (iii) of Lemma 3.1.
In the case where the functions φ n 's are of the form (1.2) we need a condition on the functions K n 's guaranteeing that (4.1) holds. Proof. By Lemma 4.1 we have P : Then the Minkowski inequality for integrals and condition (4.1) imply that The Banach fixed point theorem completes the proof of the main part of the theorem. Assertions (i) and (ii) hold because all the sets Vol. 17 (2015) Inhomogeneous refinement type equations 517

Continuous and bounded solutions
Throughout this section we assume that g ∈ C B . In contrast to the previous section we recall that f ∈ C B is called solution of equation (1.1), if (1.1) holds for every x ∈ R m .
We begin with conditions on φ n 's under which P : Proof. Fix a function f ∈ C B . Conditions (3.1)-(3.3) imply that the function and by (5.1) we obtain for all ω n ∈ Ω n and n ∈ {1, . . . , N}. Hence for every y ∈ R m and the Lebesgue dominated convergence theorem implies that lim k→∞ P f(x k ) = P f(x). In consequence, P f ∈ C B . R. Kapica and J. Morawiec In the case where the functions φ n 's are of the form (1.2), we need a condition on the functions K n 's guaranteeing that (5.1) holds.
Remark 5.2. Let K n : Ω n → R m×m and M n : Ω n → R m be Lebesgue measurable functions. If det K n ∈ L 1 (Ω n ), then the function φ n : R m × Ω n → R m given by (1. We end this paper with the following immediate consequence of Theorem 5.3 and Remarks 3.2 and 5.2. Vol. 17 (2015) Inhomogeneous refinement type equations 519 Inhomogeneous refinement type equations 13 Corollary 5.4. Assume that for every n ∈ {1, . . . , N} the functions K n : Ω n → R m×m and M n : Ω n → R m are Lebesgue measurable with det K n ∈ L 1 (Ω n ). If ∑ N n=1 ∥det K n ∥ 1 < 1, then equation (1.3) has exactly one solution f ∈ C B and for every f 0 ∈ C B the sequence (P n f 0 ) n∈N converges to f in the supremum norm. Moreover, (i) if g is of constant sign, then so is f ; (ii) if g ∈ C 0 , then f ∈ C 0 ; (iii) if there exists a closed set Z ⊂ R m such that (3.6) holds and if supp g ⊂ Z, then supp f ⊂ Z; in particular, if Z is compact, then f is compactly supported.