1,721,058 research outputs found
Generalized sampling operators with derivative samples
No full textThe generalized sampling operator is able to approximate bounded continuous functions. It is modeled on the sampling expansion for band-limited functions given by the Whittaker-Kotel'nikov-Shannon theorem. During the decades, some variations of this classical theorem have been proposed. One of them (dating back to Jagerman and Fogel and, in a more general form, to Linden and Abramson) takes into consideration also the derivative samples for the reconstruction of bandlimited functions, with a consequent benefit of a larger sampling rate compared to the Whittaker-Kotel'nikov-Shannon theorem. Motivated by this new reconstruction, we modify the generalized sampling operator including the samplings of derivatives up to a generic order to approximate non necessarily band-limited functions. One of the main features of this new operator (which we call an Hermite-type sampling operator) is the faster order of approximation. Besides the convergence and its rate, we study well-posedness, regularity, simultaneous approximation and Voronovskayatype formula. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and dat
An equivalent formulation of 0-closed sesquilinear forms
Full text linkIn 1970, McIntosh introduced the so-called 0-closed sesquilinear forms and proved a corresponding representation theorem. In this paper, we give a simple equivalent formulation of 0-closed sesquilinear forms. The main underlying idea is to consider minimal pairs of non-negative dominating forms
On some dual frames multipliers with at most countable spectra
Full text linkA dual frames multiplier is an operator consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames in a Hilbert space, respectively. In this paper we investigate the spectra of some dual frames multipliers giving, in particular, conditions to be at most countable. The contribution extends the results available in literature about the spectra of Bessel multipliers with symbol decaying to zero and of multipliers of dual Riesz bases
Sesquilinear forms associated to sequences on Hilbert spaces
Full text linkThe possibility of defining sesquilinear forms starting from one or two
sequences of elements of a Hilbert space is investigated. One can associate
operators to these forms and in particular look for conditions to apply
representation theorems of sesquilinear forms, such as Kato's theorems. The
associated operators correspond to classical frame operators or weakly-defined
multipliers in the bounded context. In general some properties of them, such as
the invertibility and the resolvent set, are related to properties of the
sesquilinear forms. As an upshot of this approach new features of sequences (or
pairs of sequences) which are semi-frames (or reproducing pairs) are obtained.Comment: 25 page
A Lebesgue-type decomposition for non-positive sesquilinear forms
Full text linkA Lebesgue-type decomposition of a (non necessarily non-negative)
sesquilinear form with respect to a non-negative one is studied. This
decomposition consists of a sum of three parts: two are dominated by an
absolutely continuous form and a singular non-negative one, respectively, and
the latter is majorized by the product of an absolutely continuous and a
singular non-negative forms. The Lebesgue decomposition of a complex measure is
given as application.Comment: 18 page
Orbits of bounded bijective operators and Gabor frames
Full text linkThis paper is a contribution to frame theory. Frames in a Hilbert space are
generalizations of orthonormal bases. In particular, Gabor frames of
, which are made of translations and modulations of one or
more windows, are often used in applications. More precisely, the paper deals
with a question posed in the last years by Christensen and Hasannasab about the
existence of overcomplete Gabor frames, with some ordering over ,
which are orbits of bounded operators on . Two classes of
overcomplete Gabor frames which cannot be ordered over and
represented by orbits of operators in are given. Some
results about operator representation are stated in a general context for
arbitrary frames, covering also certain wavelet frames.Comment: 12 page
Curves defined by a class of discrete operators: Approximation result and applications
Full text linkIn approximation theory, classical discrete operators, like generalized sampling, Szasz-Mirak'jan, Baskakov, and Bernstein operators, have been extensively studied for scalar functions. In this paper, we look at the approximation of curves by a class of discrete operators, and we exhibit graphical examples concerning several cases. The topic has useful implications about the computer graphics and the image processing: We discuss applications on the approximation and the reconstruction of curves in images
Generalized frame operator, lower semi-frames and sequences of translates
Full text linkGiven an arbitrary sequence of elements of
a Hilbert space , the operator
is defined as the operator associated to the sesquilinear form for . This operator is in
general different from the classical frame operator but possesses some
remarkable properties. For instance, is always self-adjoint in regards
to a particular space, unconditionally defined and, when is a lower
semi-frame, gives a simple expression of a dual of . The operator
and lower semi-frames are studied in the context of sequences of
integer translates of a function of . In particular, an
explicit expression of is given in this context and a characterization
of sequences of integer translates which are lower semi-frames is proved
Estimate of the spectral radii of Bessel multipliers and consequences
Full text linkBessel multipliers are operators defined from two Bessel sequences of
elements of a Hilbert space and a complex sequence, and have frame multipliers
as particular cases. In this paper an estimate of the spectral radius of a
Bessel multiplier is provided involving the cross Gram operator of the two
sequences. As an upshot, it is possible to individuate some regions of the
complex plane where the spectrum of a multiplier of dual frames is contained.Comment: 9 page
A Kato's second type representation theorem for solvable sesquilinear forms
Full text linkKato's second representation theorem is generalized to solvable sesquilinear
forms. These forms need not be non-negative nor symmetric. The representation
considered holds for a subclass of solvable forms (called hyper-solvable),
precisely for those whose domain is exactly the domain of the square root of
the modulus of the associated operator. This condition always holds for closed
semibounded forms, and it is also considered by several authors for symmetric
sign-indefinite forms. As a consequence, a one-to-one correspondence between
hyper-solvable forms and operators, which generalizes those already known, is
established.Comment: 20 page
- …
