On wavelet induced isomorphisms for joint (d, -d)-dilation wavelet and multiwavelet sets

2015 ◽  
Vol 13 (05) ◽  
pp. 1550034
Author(s):  
Pooja Singh ◽  
Dania Masood
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
Dyadic Wavelet ◽  
Wavelet Set ◽  
Almost Everywhere ◽  
Real Anal ◽  
New Construction ◽  
Fixed Point Sets

For a dyadic wavelet set W, Ionascu [A new construction of wavelet sets, Real Anal. Exchange28 (2002) 593–610] obtained a measurable self-bijection on the interval [0, 1), called the wavelet induced isomorphism of [0, 1), denoted by [Formula: see text]. Extending the result for a d-dilation wavelet set, we characterize a joint (d, -d)-dilation wavelet set, where |d| is an integer greater than 1, in terms of wavelet induced isomorphisms. Its analogue for a joint (d, -d)-dilation multiwavelet set has also been provided. In addition, denoting by [Formula: see text], the wavelet induced isomorphism associated with a d-dilation wavelet set W, we show that for a joint (d, -d)-dilation wavelet set W, the measures of the fixed point sets of [Formula: see text] and [Formula: see text] are equal almost everywhere.

ON WAVELET INDUCED ISOMORPHISMS

2010 ◽  
Vol 08 (03) ◽  
pp. 359-371 ◽  
Author(s):  
DIVYA SINGH
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
One Dimensional ◽  
Dyadic Wavelet ◽  
Wavelet Set ◽  
Fixed Point Sets

While constructing a dyadic wavelet set through an approach which is purely set-theoretic, Ionascu observed that a dyadic one-dimensional wavelet set W gives rise to a specific measurable, bijective, piecewise increasing selfmap [Formula: see text] on [0, 1) and termed it to be a wavelet induced isomorphism. Further, he found that such maps provide wavelet sets which, in turn, characterize wavelet sets. In this paper, we consider two-interval, three-interval and symmetric four-interval wavelet sets and determine their wavelet induced isomorphisms. Also, fixed point sets of [Formula: see text] are determined for these wavelet sets.

On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms

2016 ◽  
Vol 14 (03) ◽  
pp. 1650016
Author(s):  
Swati Srivastava ◽  
G. C. S. Yadav
Keyword(s):  
Fixed Point ◽  
Point Sets ◽  
Wavelet Sets ◽  
Wavelet Set ◽  
Fixed Point Sets

In this paper, we study fixed point sets of wavelet induced isomorphisms for symmetric wavelet sets. Also, introducing the notion of frame induced monomorphism for a frame wavelet set, as a generalization of the wavelet induced isomorphism for a wavelet set, we provide a construction of frame wavelet sets in [Formula: see text]. We also study fixed point sets of these maps.

On wavelet induced isomorphisms for reducing subspaces

2016 ◽  
Vol 14 (03) ◽  
pp. 1650009 ◽  
Author(s):  
Swati Srivastava ◽  
G. C. S. Yadav
Keyword(s):  
Hardy Space ◽  
Wavelet Sets ◽  
Reducing Subspace ◽  
Wavelet Set ◽  
Michigan Math ◽  
Reducing Subspaces ◽  
Real Anal ◽  
New Construction ◽  
Classical Hardy Space

In this paper, we adapt the notion of a wavelet induced isomorphism of [Formula: see text] associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an [Formula: see text]-wavelet set, where [Formula: see text] is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric [Formula: see text]-wavelet sets. Examples when [Formula: see text] is the classical Hardy space are also considered.

Fixed point sets for abstract volterra operators on fréchet spaces

2000 ◽  
Vol 76 (1-2) ◽  
pp. 131-152 ◽  
Author(s):  
Dana M. Bedivan ◽  
Donal O′Regan
Keyword(s):  
Fixed Point ◽  
Fréchet Spaces ◽  
Point Sets ◽  
Volterra Operators ◽  
Fixed Point Sets

scholarly journals Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

Biconnected Multifunctions of Trees which have an end Point as Fixed Point or Coincidence

1971 ◽  
Vol 23 (3) ◽  
pp. 461-467 ◽  
Author(s):  
Helga Schirmer
Keyword(s):  
Fixed Point ◽  
Monotone Mapping ◽  
Monotone Mappings ◽  
Point Sets ◽  
End Point ◽  
Continuous Use ◽  
Fixed Point Sets

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].

Sign in / Sign up

Export Citation Format

Share Document