We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on ...
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We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fre acute accent chet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.
For locally compact Hausdorff spaces X and Y, and function algebras A and B on X and Y, respectively, surjections T : A -> B satisfying norm multiplicative condition parallel to T f T g parallel to(Y) = parallel to...
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For locally compact Hausdorff spaces X and Y, and function algebras A and B on X and Y, respectively, surjections T : A -> B satisfying norm multiplicative condition parallel to T f T g parallel to(Y) = parallel to f g parallel to(X), f, g is an element of A, with respect to the supremum norms, and those satisfying parallel to vertical bar T f vertical bar + vertical bar T g vertical bar parallel to(Y) = parallel to vertical bar f vertical bar + vertical bar g vertical bar parallel to(X) have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups A and B of C-0(X) and C-0(Y), respectively, and study surjections T from A onto B satisfying the norm condition rho(T f, T g) = rho (f, g), f, g is an element of A, for some classes of two variable positive functions rho. It is shown that such a map T is also a composition in modulus map.
It is well known that we can use wavelets to characterize various function spaces, for example, Lebesgue, Sobolev, and Besov spaces, and get equivalent norms with wavelet coefficients. However, we cannot determine whe...
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It is well known that we can use wavelets to characterize various function spaces, for example, Lebesgue, Sobolev, and Besov spaces, and get equivalent norms with wavelet coefficients. However, we cannot determine whether a function is in these spaces by looking only at the wavelet coefficients since the constant function is orthogonal to all wavelets. In this paper, we close the gap by investigating the convergence of wavelet series.
In this paper we consider the topological structure problem for the space C(X) of all composition operators on a space X of holomorphic functions over the unit disc, where X is one of the following function spaces: th...
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In this paper we consider the topological structure problem for the space C(X) of all composition operators on a space X of holomorphic functions over the unit disc, where X is one of the following function spaces: the Hardy space H2, Bergman spaces Ap alpha, the space H infinity, weighted Banach spaces Hv with sup-norm, the classical Bloch space B, and weighted Bloch spaces Bv. In particular, we give a necessary and sufficient condition for two linear fractional composition operators to be in the same component of C(X). A characterization of isolated such composition operators in C(X) is also established.(c) 2023 Elsevier Inc. All rights reserved.
Based on the theory of fractal functions, in previous papers, the first author introduced fractal versions of functions in GP-spaces, associated fractal operator and some related notions. More recently, it has been re...
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Based on the theory of fractal functions, in previous papers, the first author introduced fractal versions of functions in GP-spaces, associated fractal operator and some related notions. More recently, it has been realized that the fractalization of a Lebesgue integrable function can be viewed as an internal binary operation, termed fractal convolution, of the germ function and a parameter map. In the current note, we continue to study this fractal convolution with a different viewpoint in mind. In particular, we consider both left and right partial fractal convolution operators on GP-spaces. As an application, we obtain bases and frames consisting of fractal functions by exploring fractal convolutions and their intriguing links with the perturbation theory of Schauder bases and frames. Thus, the theory of fractal functions and theory of bases and frames seem to come together very nicely via fractal convolution. Also established along the way are results involving bases and frames obtained by using partial fractal convolutions with the null function.
Let Y be an absolute neighbourhood retract (ANR) for the class of metric spaces and let X be a topological space. Let Y-X denote the space of continuous maps from X to Y equipped with the compact open topology. We sho...
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Let Y be an absolute neighbourhood retract (ANR) for the class of metric spaces and let X be a topological space. Let Y-X denote the space of continuous maps from X to Y equipped with the compact open topology. We show that if X is a compactly generated Tychonoff space and Y is not discrete, then Y-X is an ANR for metric spaces if and only if X is hemicompact and Y-X has the homotopy type of a CW complex.
We study the sequence spaces and the spaces of functions defined on interval [0, 1] in this paper. By a new summation method of sequences, we find out some new sequence spaces that are interpolating into spaces betwee...
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We study the sequence spaces and the spaces of functions defined on interval [0, 1] in this paper. By a new summation method of sequences, we find out some new sequence spaces that are interpolating into spaces between l(p) and l(q) and function spaces that are interpolating into the spaces between the polynomial space P[0, 1] and C-infinity[0, 1]. We prove that these spaces of sequences and functions are Banach spaces.
The additivity of D-property is studied on t-metrizable spaces and certain function spaces. It is shown that a space of countable tightness is a D-space provided that it is the union of finitely many t-metrizable subs...
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The additivity of D-property is studied on t-metrizable spaces and certain function spaces. It is shown that a space of countable tightness is a D-space provided that it is the union of finitely many t-metrizable subspaces, or function spaces C-p(X-i) where each X-i is Lindelof Sigma.
By combining frequency-uniform decomposition with L-p (l(q)), we introduce a new class of function spaces (denoted by X-p,X-q,(s)). Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-...
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By combining frequency-uniform decomposition with L-p (l(q)), we introduce a new class of function spaces (denoted by X-p,X-q,(s)). Moreover, we study the Cauchy problem for the generalized NLS equations and Ginzburg-Landau equations in L-r (0, T;X-p,1(s)).
Let (Omega, I pound, mu) be a finite atomless measure space, and let E be an ideal of L (0)(mu) such that . We study absolutely continuous linear operators from E to a locally convex Hausdorff space . Moreover, we exa...
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Let (Omega, I pound, mu) be a finite atomless measure space, and let E be an ideal of L (0)(mu) such that . We study absolutely continuous linear operators from E to a locally convex Hausdorff space . Moreover, we examine the relationships between mu-absolutely continuous vector measures m : I pound -> X and the corresponding integration operators T (m) : L (a)(mu) -> X. In particular, we characterize relatively compact sets in ca (mu) (I pound, X) (= the space of all mu-absolutely continuous measures m : I pound -> X) for the topology of simple convergence in terms of the topological properties of the corresponding set of absolutely continuous operators. We derive a generalized Vitali-Hahn-Saks type theorem for absolutely continuous operators T : L (a)(mu) -> X.
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