New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

Castro, L. P.; Minh, L. T.; Tuan, Nguyen Minh

doi:10.1007/s00009-017-1063-y

Cited by 53 publications

(29 citation statements)

References 24 publications

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“…Proof By using the Cauchy inequality, we have

\int_{double-struckR} | (\tilde{f} #_{Ω_{d}^{a}} \tilde{ψ)} (t) {true|}^{2} d t \leq {(\int_{double-struckR} | f (ω) | d ω)}^{2} (\int_{double-struckR} | ψ (x) {false|}^{2} d x) < \infty .

This shows that

\overset{f}{true} #_{{normalΩ}_{d}^{a}} \overset{ψ}{true} \in L^{2} false(double-struckR false)

. Moreover, by using the admissibility condition for

\overset{f}{true} #_{{normalΩ}_{d}^{a}} \overset{ψ}{true}

and from, we have

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 & eqnarray-2 & eqnarray-3 \int_{double-struckR} \frac{{|{scriptQ}_{d, e}^{a, b, c} e^{- a {(\cdot)}^{2} - i d (\cdot)} (\tilde{f} #_{{normalΩ}_{d}^{a}} \tilde{ψ}) (ω)|}^{2}}{| ω |} d ω \\ eqnarray-1 & eqnarray-2 & eqnarray-3 = \int_{double-struckR} |\sqrt{\frac{2 π i}{b}} e^{- i c ω^{2} - i e ω} ({scriptQ}_{d, e}^{a, b, ...})| \end{matrix}

…”

Section: Quadratic‐phase Fourier Wavelet Transformmentioning

confidence: 88%

“…In past decades, many different types of integral transforms such as Fourier, fractional Fourier, and linear canonical transforms are discussed in time‐frequency analysis. Now, more recently, the quadratic‐phase Fourier transform (QPFT) is defined as a generalization of several integral transforms whose kernel in exponential form. With minor modification in Castro et al, the QPFT with five real parameters a , b , c , d , e of a function

f \in L^{1} false(double-struckR false)

f \in L^{2} false(double-struckR false)

is denoted by

{scriptQ}_{d, e}^{a, b, c} f

and defined as

((), Q_{d, e}^{a, b, c} f) false(ω false) = \overset{f}{true} false(ω false) = \int_{R} {scriptK}_{d, e}^{a, b, c} false(ω, t false) f false(t false) d t,

where

{scriptK}_{d, e}^{a, b, c} false(ω, t false) = {scriptK}_{e, d}^{c, b, a} false(t, ω false) = \sqrt{\frac{b}{2 π i}} e^{i Ω_{d, e}^{a, b, c} false(ω, t false)}

and

{normalΩ}_{d, e}^{a, b, c} false(ω, t false) = a t^{2} + b t ω + c ω^{2} + d t + e ω

which is known as the quadratic‐phase function.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The quadratic‐phase Fourier wavelet transform

Prasad

Sharma

2019

Math Methods in App Sciences

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In this paper, we define the quadratic-phase Fourier wavelet transform (QPFWT) and discuss its basic properties including convolution for QPFWT. Further, inversion formula and the Parseval relation of QPFWT are also discussed.Continuity of QPFWT on some function spaces are studied. Moreover, some applications of quadratic-phase Fourier transform (QPFT) to solve the boundary value problems of generalized partial differential equations. KEYWORDS convolution, generalized Sobolev type space, quadratic-phase Fourier transform, quadratic-phase Fourier wavelet transform, wavelet transform MSC CLASSIFICATION

show abstract

“…Proof By using the Cauchy inequality, we have

\int_{double-struckR} | (\tilde{f} #_{Ω_{d}^{a}} \tilde{ψ)} (t) {true|}^{2} d t \leq {(\int_{double-struckR} | f (ω) | d ω)}^{2} (\int_{double-struckR} | ψ (x) {false|}^{2} d x) < \infty .

This shows that

\overset{f}{true} #_{{normalΩ}_{d}^{a}} \overset{ψ}{true} \in L^{2} false(double-struckR false)

. Moreover, by using the admissibility condition for

\overset{f}{true} #_{{normalΩ}_{d}^{a}} \overset{ψ}{true}

and from, we have

\begin{matrix} eqnarrayleft center right \\ eqnarray-1 & eqnarray-2 & eqnarray-3 \int_{double-struckR} \frac{{|{scriptQ}_{d, e}^{a, b, c} e^{- a {(\cdot)}^{2} - i d (\cdot)} (\tilde{f} #_{{normalΩ}_{d}^{a}} \tilde{ψ}) (ω)|}^{2}}{| ω |} d ω \\ eqnarray-1 & eqnarray-2 & eqnarray-3 = \int_{double-struckR} |\sqrt{\frac{2 π i}{b}} e^{- i c ω^{2} - i e ω} ({scriptQ}_{d, e}^{a, b, ...})| \end{matrix}

…”

Section: Quadratic‐phase Fourier Wavelet Transformmentioning

confidence: 88%

f \in L^{1} false(double-struckR false)

f \in L^{2} false(double-struckR false)

is denoted by

{scriptQ}_{d, e}^{a, b, c} f

and defined as

((), Q_{d, e}^{a, b, c} f) false(ω false) = \overset{f}{true} false(ω false) = \int_{R} {scriptK}_{d, e}^{a, b, c} false(ω, t false) f false(t false) d t,

where

{scriptK}_{d, e}^{a, b, c} false(ω, t false) = {scriptK}_{e, d}^{c, b, a} false(t, ω false) = \sqrt{\frac{b}{2 π i}} e^{i Ω_{d, e}^{a, b, c} false(ω, t false)}

and

{normalΩ}_{d, e}^{a, b, c} false(ω, t false) = a t^{2} + b t ω + c ω^{2} + d t + e ω

which is known as the quadratic‐phase function.…”

Section: Introductionmentioning

confidence: 99%

The quadratic‐phase Fourier wavelet transform

Prasad

Sharma

2019

Math Methods in App Sciences

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“…As far as it concerns convolutions and their consequences, a huge amount of works could also be pointed out due to both theoretical and practical perspectives. Anyway, for this purpose, as examples, we simply refer the interested reader to [4,5,11,12,13,16] and to the references therein.…”

Section: Introductionmentioning

confidence: 99%

New sampling theorem and multiplicative filtering in the FRFT domain

Anh

Castro

Thao

et al. 2019

SIViP

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Having in consideration a fractional convolution associated with the fractional Fourier transform (FRFT), we propose a novel reconstruction formula for bandlimited signals in the FRFT domain without using the classical Shannon theorem. This may be considered the main contribution of this work, and numerical experiments are implemented to demonstrate the effectiveness of the proposed sampling theorem. As a second goal, we also look for the designing of multiplicative filters. Indeed, we also convert the multiplicative filtering in FRFT domain to the time domain, which can be realized by Fast Fourier transform. Two concrete examples are included where the use of the present results is illustrated.

show abstract

“…Among the integral operators we may classify a great variety of different types of operators, cf. [1,2,3,4,5,6,7,8,9,10]. In some cases, there are convolutions somehow associated with the integral operators and allow the consideration of consequent convolution type equations.…”

Section: Introduction and Basic Propertiesmentioning

confidence: 99%

“…which satisfy the following identities: (5) (see [11,12]). So, we can write our operator in terms of these projectors as follows:…”

Section: Introduction and Basic Propertiesmentioning

confidence: 99%

On integral operators and equations generated by cosine and sine Fourier transforms

Castro¹,

Guerra²,

Tuan³

2018

AIP Conference Proceedings

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In this paper, we study some properties of a class of integral operators that depends on the cosine and sine Fourier transforms. In particular, we will exhibit properties related with their invertibility, the spectrum, Parseval type identities and involutions. Moreover, a new convolution will be proposed and consequent integral equations will be also studied in detail. Namely, we will characterize the solvability of two integral equations which are associated with the integral operator under study. Moreover, under appropriate conditions, the unique solutions of those two equations are also obtained in a constructive way. n 2 R n sin(xy) f (y)dy. We shall also recall the concept of algebraic operators. For this matter, let X be a linear space over the complex field C, and let L(X) be the set of all linear operators with domain and range in X. Definition 1 (see [11, 12]) An operator K ∈ L(X) is said to be algebraic if there exists a normed (non-zero) polynomial P K (t) = t m + α 1 t m−1 + • • • + α m−1 t + α m , α j ∈ C, j = 1, 2,. .. , m such that P K (K) = 0 on X.

show abstract

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

Cited by 53 publications

References 24 publications

The quadratic‐phase Fourier wavelet transform

The quadratic‐phase Fourier wavelet transform

New sampling theorem and multiplicative filtering in the FRFT domain

On integral operators and equations generated by cosine and sine Fourier transforms

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