scholarly journals Spectral Functions for the Vector-Valued Fourier Transform

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Spectral Function ◽  
Vector Function ◽  
Finite Interval ◽  
Spectral Functions ◽  
The Matrix ◽  
The Fourier Transform ◽  
Vector Valued

A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

NOTE ON THE VLASOV EQUATION FOR PLASMAS

1963 ◽  
Vol 41 (12) ◽  
pp. 1960-1966 ◽  
Author(s):  
Ta-You Wu ◽  
M. K. Sundaresan
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Vlasov Equation ◽  
Hermite Polynomials ◽  
Infinite Matrix ◽  
Mean Square ◽  
Initial Value ◽  
Positive Ions ◽  
The Mean ◽  
The Fourier Transform

The linearized Vlasov equation is solved as an initial value problem by expanding (the Fourier components of) the distribution function in a series of Hermite polynomials in the momentum, with coefficients which are functions of time. The spectrum of frequencies is given by the eigenvalues of an infinite matrix. All the frequencies ω are real, extending from small values of order ω2 = k2(u22), where (u22) is the mean square velocity of the positive ions (of mass M), to [Formula: see text], where ω1, (u12) are the plasma frequency and mean square velocity of the electrons (of mass m). The classic work of Landau solves the Vlasov equation for (the Fourier transform of) the potential for which he obtains the "damping", whereas Van Kampen and the present writers solve the equation for (the Fourier transform of) the distribution function itself. While the present work gives results equivalent to those of Van Kampen, the method is simpler and in fact elementary.

scholarly journals Controlled Release of Agrochemicals Intercalated into Montmorillonite Interlayer Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Harrison Wanyika
Keyword(s):  
Fourier Transform ◽  
Controlled Release ◽  
Interlayer Space ◽  
Rotary Evaporation ◽  
Surface Areas ◽  
Space Reduction ◽  
Immersion Technique ◽  
The Matrix ◽  
The Fourier Transform ◽  
Mmt Clay

Periodic application of agrochemicals has led to high cost of production and serious environmental pollution. In this study, the ability of montmorillonite (MMT) clay to act as a controlled release carrier for model agrochemical molecules has been investigated. Urea was loaded into MMT by a simple immersion technique while loading of metalaxyl was achieved by a rotary evaporation method. The successful incorporation of the agrochemicals into the interlayer space of MMT was confirmed by several techniques, such as, significant expansion of the interlayer space, reduction of Barrett-Joyner-Halenda (BJH) pore volumes and Brunauer-Emmett-Teller (BET) surface areas, and appearance of urea and metalaxyl characteristic bands on the Fourier-transform infrared spectra of the urea loaded montmorillonite (UMMT) and metalaxyl loaded montmorillonite (RMMT) complexes. Controlled release of the trapped molecules from the matrix was done in water and in the soil. The results reveal slow and sustained release behaviour for UMMT for a period of 10 days in soil. For a period of 30 days, MMT delayed the release of metalaxyl in soil by more than 6 times. It is evident that MMT could be used to improve the efficiency of urea and metalaxyl delivery in the soil.

scholarly journals Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

2005 ◽  
Vol 42 (03) ◽  
pp. 620-631
Author(s):  
M. Möhle
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Probability Measure ◽  
Integral Representation ◽  
Convergence Result ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Convergence Results ◽  
The Fourier Transform ◽  
Related Paper

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

AUTOMATIC THREE‐DIMENSIONAL MODELING FOR THE INTERPRETATION OF GRAVITY OR MAGNETIC ANOMALIES

Geophysics ◽  
1975 ◽  
Vol 40 (6) ◽  
pp. 1014-1034 ◽  
Author(s):  
A. Gerard ◽  
N. Debeglia
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Iterative Method ◽  
Three Dimensional ◽  
Magnetic Anomalies ◽  
Average Depth ◽  
Three Dimensional Modeling ◽  
Dimensional Modeling ◽  
The Fourier Transform ◽  
Homogeneous Media

Transformation of gravity or magnetic anomaly maps into isodepth maps of a surface separating two homogeneous media may be accomplished by (1) systematically estimating an average depth and density or magnetization contrast for the surface and (2) using an iterative method to adjust local depths compared to the average depth of the surface. Average depth, density or magnetization contrast, and iterative adjustment of local depths are determined using the Fourier transform of the field to be interpreted and that of the field generated by an equivalent surface. This leads us to propose a method of estimating the average depth of the sources and a distribution function for the depths and then a complete and very economical algorithm for the calculation of the corresponding equivalent surface.

scholarly journals Closure of dilates of shift-invariant subspaces

2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Moisés Soto-Bajo
Keyword(s):  
Fourier Transform ◽  
Spectral Function ◽  
Invariant Subspace ◽  
Invariant Subspaces ◽  
General Setting ◽  
Tight Frame ◽  
Local Behaviour ◽  
Approximate Continuity ◽  
Completeness Property ◽  
The Fourier Transform

AbstractLet V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.

scholarly journals Dynamic RCS Estimation according to Drone Movement Using the MoM and Far-Field Approximation

2021 ◽  
Vol 21 (4) ◽  
pp. 322-328
Author(s):  
Dong-Yeob Lee ◽  
Jae-In Lee ◽  
Dong-Wook Seo
Keyword(s):  
Fourier Transform ◽  
Cross Section ◽  
Method Of Moments ◽  
Iterative Process ◽  
Rotation Angle ◽  
Field Approximation ◽  
Far Field ◽  
Impedance Matrix ◽  
The Matrix ◽  
The Fourier Transform

Micro-Doppler signatures from the rotating propellers of a drone can be utilized to distinguish the drone from clutter or airborne organisms with similar radar cross section (RCS) levels, such as birds and bats. To obtain the micro-Doppler signatures of a drone, calculation or measurement of the electric field scattered from the rotating propellers is essential. In this paper, using the relative angle concept and far-field approximation, we propose a way to rapidly estimate the dynamic RCS of a drone with several propellers according to its movement. In addition, based on the fact that the shape of the propeller does not change even if it rotates, we construct an impedance matrix only once and apply the matrix to the method of moments instead of the iterative process of calculating the impedance matrix and inverse matrix for each rotation angle of the propeller. Finally, by using the Fourier transform of the results from the proposed method, the rotation frequencies of the propellers according to the movement of the drone can be obtained.

scholarly journals Application of FTIR Method for the Assessment of Immobilization of Active Substances in the Matrix of Biomedical Materials

Materials ◽  
2019 ◽  
Vol 12 (18) ◽  
pp. 2972 ◽  
Author(s):  
Dorota Kowalczuk ◽  
Monika Pitucha
Keyword(s):  
Fourier Transform ◽  
Fourier Transform Infrared ◽  
Spectroscopic Technique ◽  
Modification Process ◽  
The Matrix ◽  
Ideal Tool ◽  
Reflectance Ftir ◽  
The Fourier Transform ◽  
Specific Reactions ◽  
Active Substances

Background: The purpose of the study was to demonstrate the usefulness of the Fourier transform infrared spectroscopy (FTIR) method for the evaluation of the modification process of biomaterials with the participation of active substances. Methods: Modified catheter samples were prepared by activating the matrix with an acid, iodine, or bromine, and then immobilizing the active molecules. To carry out the modification process, the Fourier transform infrared-attenuated total reflectance (FTIR-ATR) method was used. Results: FTIR analysis indicated the presence of the immobilized substances in the catheter matrix and site-specific reactions. Conclusion: We surmise that the infrared spectroscopic technique is an ideal tool for the assessment of the drug immobilization and the changes occurring in the course of the modification process.

scholarly journals REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION

2020 ◽  
Vol 8 (2) ◽  
pp. 103-113
Author(s):  
H. Lopushanska ◽  
A. Lopushansky
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Fourier Transform ◽  
Approximate Solution ◽  
Vector Function ◽  
Unique Solvability ◽  
Fractional Derivatives ◽  
Direct And Inverse Problems ◽  
The Cauchy Problem ◽  
The Fourier Transform

Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

scholarly journals The Fourier transform of vector-valued functions

1985 ◽  
Vol 26 (2) ◽  
pp. 181-186
Author(s):  
Susumu Okada
Keyword(s):  
Fourier Transform ◽  
Integrable Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bounded Continuous Function ◽  
The Fourier Transform ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Convergence In Mean

For each natural number n, let un(x)=(1—cos nx)/πnx2(xɛℝ). It is well–known that a bounded continuous function f on the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined byform a Cauchy sequence in the space L1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the space L1(ℝ) is complete with respect to convergence in mean.

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