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Patrik Wahlberg

Publications and source records attributed to Patrik Wahlberg.

At least 19 recordsLinked to original sources
arXiv2026

Filtering of second order generalized stochastic processes corrupted by additive noise

We treat the optimal linear filtering problem for a sum of two second order uncorrelated generalized stochastic processes. This is an operator equation involving covariance operators. We study both the wide-sense stationary case and the non-stationary case. In the former case the equation simplifies into a convolution equation. The solution is the Radon--Nikodym derivative between non-negative tempered Radon measures, for signal and signal plus noise respectively, in the frequency domain. In the non-stationary case we work with pseudodifferential operators with symbols in Sjöstrand modulation spaces which admits the use of its spectral invariance properties.

math.FA↗
arXiv2026

The filter of singularities in global anisotropic microlocal analysis

We define a filter of time-frequency anisotropic global singularities of phase space for tempered distributions. The filter contains information from the corresponding anisotropic Gabor wave front set and admits propagation results for the Cauchy problem for certain linear evolution equations of Schrödinger type that generalize the harmonic oscillator.

math.AP↗
arXiv2025

The Wigner distribution of Gaussian tempered generalized stochastic processes

We define the Wigner distribution of a tempered generalized stochastic process that is complex-valued symmetric Gaussian. This gives a time-frequency generalized stochastic process defined on the phase space. We study its covariance and our main result is a formula for the Weyl symbol of the covariance operator, expressed in terms of the Weyl symbol of the covariance operator of the original generalized stochastic process.

math.PR↗
arXiv2025

Propagation of singularities for anharmonic Schrödinger equations

We consider evolution equations for two classes of generalized anharmonic oscillators and the associated initial value problem in the space of tempered distributions. We prove that the Cauchy problem is well posed in anisotropic Shubin--Sobolev modulation spaces of Hilbert type, and we investigate propagation of suitable notions of singularities.

math.AP↗
arXiv2024

Polynomially oscillatory multipliers on Gelfand-Shilov spaces

We study continuity of the multiplier operator $e^{i q}$ acting on Gelfand--Shilov spaces, where $q$ is a polynomial on $\mathbf R^d$ of degree at least two with real coefficients. In the parameter quadrant for the spaces we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well-posedness for linear evolution equations that generalize the Schrödinger equation for the free particle.

math.FA↗
arXiv2024

Propagation of anisotropic Gabor wave front sets

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

math.AP↗
arXiv2024

Propagation of anisotropic Gabor singularities for Schrödinger type equations

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity $a(λx, λ^σξ) = λ^{1+σ} a(x,ξ)$ for $λ> 0$ where $σ> 0$ is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin--Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.

math.AP↗
arXiv2023

Semigroups for quadratic evolution equations acting on Shubin-Sobolev and Gelfand-Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin--Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand--Shilov spaces and their dual ultradistribution spaces.

math.AP↗
arXiv2023

Microlocal analysis for Gelfand--Shilov spaces

We introduce an anisotropic global wave front set of Gelfand--Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand--Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

math.AP↗
arXiv2022

Anisotropic global microlocal analysis for tempered distributions

We study an anisotropic version of the Shubin calculus of pseudodifferential operators on $\mathbf R^d$. Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin's isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial phase function.

math.AP↗
arXiv2022

Propagation of anisotropic Gelfand-Shilov wave front sets

We show a result on propagation of the anisotropic Gelfand--Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand--Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand--Shilov wave front set of the Schwartz kernel of the operator is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.

math.AP↗
arXiv2021

Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

We study the link between pseudo-differential operators and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of pseudo-differential operators of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.

math.FA↗
arXiv2019

Lagrangian distributions and Fourier integral operators with quadratic phase functions and Shubin amplitudes

We study Fourier integral operators with Shubin amplitudes and quadratic phase functions associated to twisted graph Lagrangians with respect to symplectic matrices. We factorize such an operator as the composition of a Weyl pseudodifferential operator and a metaplectic operator and derive a characterization of its Schwartz kernel in terms of phase space estimates. Extending the conormal distributions in the Shubin calculus, we define an adapted notion of Lagrangian tempered distribution. We show that the kernels of Fourier integral operators are identical to Lagrangian distributions with respect to twisted graph Lagrangians.

math.AP↗
arXiv2019

Shubin type Fourier integral operators and evolution equations

We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin's class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.

math.AP↗
arXiv2018

The Weyl product on quasi-Banach modulation spaces

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

math.FA↗
arXiv2017

Conormal distributions in the Shubin calculus of pseudodifferential operators

We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.

math-ph↗
arXiv2016

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

We study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. We point out that the singular space associated to the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.

math.AP↗