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2021 International Aegean Conference on Electrical Machines and Power Electronics, ACEMP 2021 and 2021 International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2021 -- 2 September 2021 through 3 September 2021 -- -- 173666, In this paper, the operating point of variable flux reluctance motors (VFRMs) using voltage and current limits in the field weakening region is investigated. The voltage and current limits of VFRM, which depend on the current capacity of the inverter and the voltage limit of the inverter, are investigated by using MATLAB/Simulink. The d-axis and q-axis currents are represented using a mathematical model of VFRM. According to the analyses, the control algorithm for VFRMs is investigated in the field weakening region. VFRM can properly operate in the constant power region with this method. The conventional inverter is used to supply required d-axis and q-axis currents to armature windings with the space vector pulse width modulation (SVPWM) technique. The DC/DC converter is used to control field windings. For the validation of the investigated control algorithm, 12 slots, 10 poles VFRM is implemented in MATLAB/Simulink. The simulation results show that the investigated field weakening method for VFRM controls the motor in constant power region and extends the speed range. © 2021 IEEE., Türkiye Bilimsel ve Teknolojik Araştirma Kurumu, TÜBITAK: 218E057, ACKNOWLEDGMENT This work is being supported by TUBITAK (The Scientific and Technological Research Council of Turkey) under contract number 218E057.

The Wigner distribution is a milestone of Time-frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kind of modifications have been proposed. This contributions fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As main result, we provide a characterization for the Cohen's class. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role., 37 pages, 1 figure. To appear in Analysis and Applications

We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices, significantly extending the original theory introduced by Gr\"{o}chenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured., Comment: 69 pages

In this chapter, we exhibit recent advances in signal analysis via time–frequency distributions. New members of the Cohen class, generalizing the Wigner distribution, reveal to be effective in damping artefacts of some signals. We will survey their main properties and drawbacks and present open problems related to such phenomena.

In the first part of the paper we provide a survey of recent results concerning the problem of pointwise convergence of integral kernels in Feynman path integrals, obtained by means of time-frequency analysis techniques. We then focus on exceptional times, where the previous results do not hold, and we show that weaker forms of convergence still occur. In conclusion we offer some clues about possible physical interpretation of exceptional times.

Time-frequency analysis have played a crucial role in the development of localization operators in the last 20 years. We present its applications to the study of boundedness and Schatten Class property for such operators. In particular, new sufficient conditions for such operators to belong to the Schatten-von Neumann Class \(S_p(L^2(\mathbb {R}^d))\), 0 < p < 1, are exhibited. As a byproduct, sharp continuity results for the Wigner distribution are also presented.

We indicate how to construct a family of modulation function spaces that have a scaling symmetry. We also illustrate the behavior of the Schr\"odinger multiplier on such function spaces., Comment: 12 pages. To appear in Appl. Comput. Harmon. Anal

We consider time-frequency localization operators $A_a^{\varphi_1,\varphi_2}$ with symbols $a$ in the wide weighted modulation space $ M^\infty_{w}(\mathbb{R}^{2d})$, and windows $ \varphi_1, \varphi_2 $ in the Gelfand-Shilov space $\mathcal{S}^{\left(1\right)}(\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{\varphi_1,\varphi_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $ \mathcal{S}^{(\gamma)} (\mathbb{R}^{d}) $, where the parameter $\gamma \geq 1 $ is related to the growth of the considered weight. An important role is played by $\tau$-pseudodifferential operators $\mathrm{Op}_\tau(\sigma)$. In that direction we show convenient continuity properties of $\mathrm{Op}_\tau(\sigma)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\mathrm{Op}_\tau(\sigma)$ when the symbol $\sigma$ belongs to a modulation space with appropriately chosen weight functions. As a tool we also prove new convolution relations for (quasi-)Banach weighted modulation spaces., Comment: 29 pages

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

We study the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^\beta$, $\beta>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^\beta}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^{\beta}$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^\beta$ is the bottom of the spectrum of $H^\beta$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$., Comment: 18 pages

We give a characterization of Gelfand-Shilov type spaces of test functions and their dual spaces of tempered ultradistributions by the means of Wilson bases of exponential decay. We offer two different proofs, and extend known results to the Roumieu case., Comment: 20 pages

This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces., 11 pages; added reference [1]

Born-Jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by Born and Jordan in 1925. The weak definition of such operators involves the Born-Jordan distribution, first introduced by Cohen in 1966 as a member of the Cohen class. We perform a time-frequency analysis of the Cohen kernel of the Born -Jordan distribution, using modulation and Wiener amalgam spaces. We then provide sufficient and necessary conditions for Born-Jordan operators to be bounded on modulation spaces. We use modulation spaces as appropriate symbols classes., Comment: 21 pages, 1 figure

In this review we focus on the almost diagonalization of pseudodifferential operators and highlight the advantages that time-frequency techniques provide here. In particular, we retrace the steps of an insightful paper by Grochenig, who succeeded in characterizing a class of symbols previously investigated by Sjostrand by noticing that Gabor frames almost diagonalize the corresponding Weyl operators. This approach also allows to give new and more natural proofs of related results such as boundedness of operators or algebra and Wiener properties of the symbol class. Then, we discuss some recent developments on the theme, namely an extension of these results to a more general family of pseudodifferential operators and similar outcomes for a symbol class closely related to Sjostrand’s one.

This article gives explicit integral formulas for the so-called generalized metaplectic operators, i.e. Fourier integral operators (FIOs) of Schr\"odinger type, having a symplectic matrix as canonical transformation. These integrals are over specific linear subspaces of R^d, related to the d x d upper left-hand side submatrix of the underlying 2d x 2d symplectic matrix. The arguments use the integral representations for the classical metaplectic operators obtained by Morsche and Oonincx in a previous paper, algebraic properties of symplectic matrices and time-frequency tools. As an application, we give a specific integral representation for solutions to the Cauchy problem of Schr\"odinger equations with bounded perturbations for every instant time t in R, even in the so-called caustic points., Comment: 19 pages in Journal of Fourier Analysis and Applications, 2015

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$., Comment: 19 pages

Given a compact manifold $M$ with boundary $\partial M$, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to $(M,\partial M)$. The symbols of operators with boundary conditions on $\partial M$ are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator $L$ with the same boundary conditions on $\partial M$. The boundary $\partial M$ is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on $L^2(M)$ and $r$-nuclearity on $L^p(M)$ are obtained. We also describe a new addition to the Grothendieck-Lidskii formula in this setting. Examples and applications are given to operators on $M=[0,1]^n$ with non-periodic boundary conditions, and of operators with non-local boundary conditions., 27 pages; the setting of boundary value problems in Rn in the previous version has been now extended to more general compact manifolds with boundary

In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1410.4687

This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schr\"odinger type equations. Indeed, continuity results of both Schr\"odinger propagators and their asymptotic solutions are obtained on $\hbar$-dependent Banach spaces, the semi-classical version of the well-known modulation spaces. Moreover, their operator norm is controlled by a constant independent of the Planck's constant $\hbar$. The main tool in our investigation is the joint application of standard approximation techniques from semi-classical analysis and a generalized version of Gabor frames, dependent of the parameter $\hbar$. Continuity properties of more general Fourier integral operators (FIOs) and their sparsity are also investigated., Comment: 23 pages

We study the Cauchy problem for an incompressible magneto-hydrodynamics (MHD) system in the modulation space M q , σ s ( R n ) ( n ⩾ 2 ) with initial data ( u 0 , b 0 ) ∈ P M q , σ s ( R n ) × P M q , σ s ( R n ) . We prove that this problem is locally well-posed in such a space when 1 ⩽ q ⩽ ∞ , 1 ⩽ σ ∞ and n ( σ − 1 ) σ − 1 ⩽ s , and globally well-posed when 1 ⩽ q ⩽ n , n n − 1 ⩽ σ ∞ and max { n ( σ − 1 ) σ − 1 , n ( σ − 2 ) σ } s n ( σ − 1 ) σ .

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation u t = ( ν + i ) Δ u + λ → 1 ⋅ ∇ ( | u | 2 u ) + ( λ → 2 ⋅ ∇ u ) | u | 2 + α | u | 2 δ u , where δ ∈ N , λ → 1 , λ → 2 are complex constant vectors, ν ∈ [ 0 , 1 ] , α ∈ C . For n ≥ 3 , we show that it is uniformly global well posed for all ν ∈ [ 0 , 1 ] if initial data u 0 in modulation space M 2 , 1 s and Sobolev spaces H s + n / 2 ( s > 3 ) and ‖ u 0 ‖ L 2 is small enough. Moreover, we show that its solution will converge to that of the derivative Schrodinger equation in C ( 0 , T ; L 2 ) if ν → 0 and u 0 in M 2 , 1 s or H s + n / 2 with s > 4 . For n = 2 , we obtain the local well-posedness results and inviscid limit with the Cauchy data in M 1 , 1 s ( s > 3 ) and ‖ u 0 ‖ L 1 ≪ 1 .

Fourier integral operators with sufficiently smooth phase act on the time-frequency content of functions. However time-frequency analysis has only recently been used to analyze these operators. In this paper, we show that if a Fourier integral operator has a smooth phase function and its symbol is well localized in time and frequency, then the operator is Schatten p-class for p ∈ [ 1 , 2 ] , with inclusion of the symbol in mixed modulation spaces serving as the appropriate measure of time-frequency localization. Our main results are sharp in the sense that larger mixed modulation spaces necessarily contain symbols of Fourier integral operators that are not Schatten p-class.

The relevance of modulation spaces for deformation quantization, Landau--Weyl quantization and noncommutative quantum mechanics became clear in recent work. We continue this line of research and demonstrate that $Q_s(\mathbb{R}^{2d})$ is a good class of symbols for Landau-Weyl quantization and propose that the modulation spaces $M^p_{v_s}(\mathbb{R}^{2d})$ are natural generalized Shubin classes for the Weyl calculus. This is motivated by the fact that the Shubin class $Q_s(\mathbb{R}^{2d})$ is the modulation space $M^2_{v_s}(\mathbb{R}^{2d})$. The main result gives estimates of the singular values of pseudodifferential operators with symbols in $M^p_{v_s}(\mathbb{R}^{2d})$ for the standard Weyl calculus and for the Landau--Weyl calculus.

The Cauchy problems for Navier–Stokes equations and nonlinear heat equations are studied in modulation spaces M q , σ s ( R n ) . Though the case of the derivative index s = 0 has been treated in our previous work, the case s ≠ 0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of M q , σ s ( R n ) for the existence of local and global solutions for initial data u 0 ∈ M q , σ s ( R n ) .

It is known that Fourier integral operators arising when solving Schrodinger-type operators are bounded on the modulation spaces ℳp,q, for 1≤p= q≤∞, provided their symbols belong to the Sjostrand class M∞,1. However, they generally fail to be bounded on ℳp,q for p≠q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ℳp,q for p≠q, and between ℳp,q→ℳq,p, 1≤q

We discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions omega we prove that M-(omega)(p,q) is an algebra under the Weyl product if p epsilon [1, infinity] and 1

We consider the dilation property of the modulation spaces $M^{p,q}$. Let $D_\lambda:f(t)\mapsto f(\lambda t)$ be the dilation operator, and we consider the behavior of the operator norm $\|D_\lambda\|_{M^{p,q}\to M^{p,q}}$ with respect to $\lambda$. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov space, which was proved by Toft., Comment: 23 pages

The theorems of Balan, Casazza, Heil, and Landau concerning the removal of sets of positive density from frames with positive excess are extended using a more general, symmetric concept of localization of frames.

The α-modulation spaces M p , q s , α ( R d ) , α ∈ [ 0 , 1 ] , form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.

We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on L p spaces. The main tools are Gabor frames and methods from time–frequency analysis.

The boundedness of pseudodifferential operators on modulation spaces defined by the means of almost exponential weights is studied. The results are applied to symbol class with almost exponential bounds including polynomial and ultra-polynomial symbols. The Weyl correspondence is used and it is noted that the results can be transferred to the operators with appropriate anti-Wick symbols. It is proved that a class of elliptic pseudodifferential operators can be almost diagonalized by the elements of Wilson bases, and estimates for their eigenvalues are given. Furthermore, it is shown that the same can be done by using Gabor frames.

Let Mp,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p1+1/p2=1+1/p0 and 1/q1+1/q2=1/q0, then it is proved that Mp1,q1∗Mp2,q2⊂Mp0,q0. The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.

We study classes of pseudodifferential operators which are bounded on large collections of modulation spaces. The conditions on the operators are stated in terms of the L p , q estimates for the continuous Gabor transforms of their symbols. In particular, we show how these classes are related to the class of operators of Grochenig and Heil, which is bounded on all modulation spaces.

The Balian–Low theorem expresses the fact that time–frequency concentration is incompatible with non-redundancy for Gabor systems that form orthonormal or Riesz bases for L 2 ( R ) . We extend the Balian–Low theorem for Riesz bases to higher dimensions, obtaining a weak form valid for all sets of time–frequency shifts which form a lattice in R 2d , and a strong form valid for symplectic lattices in R 2d . For the orthonormal basis case, we obtain a strong form valid for general non-lattice sets which are symmetric with respect to the origin.

We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to $N=2$ of a result valid for the $N$-fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces., Comment: In this update we have corrected a sign error in the exponent in Eq. (1.7), and its consequences in Prop. 1.5 and Eq. (2.30)

A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis., Comment: 22. pages

We prove sharp estimates for the dilation operator $f(x)\longmapsto f(\lambda x)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schr\"odinger propagator on modulation spaces., Comment: 12 pages

Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NLKG equation., Comment: 13 pages

Time-frequency methods are used to study a class of Fourier Integral Operators (FIOs) whose representation using Gabor frames is proved to be very efficient. Indeed, similarly to the case of shearlets and curvelets frames [10, 35], the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty, 1}$ [33], for some Fourier multipliers [6] and metaplectic operators [14, 31]. Moreover, this paper provides the mathematical tools to numerically solving the Cauchy problem for Schr¨odinger equations using Gabor frames [17]. Finally, similar arguments can be employed to study other classes of FIOs [16].