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@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation (aka intermediate field representation). See, for instance, the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010. See in particular Sections III.A and IV.A.

@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation (aka intermediate field representation). See, for instance, the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010 and the follow up "Note on the intermediate field representation of $\phi^{2k}$ theory in zero dimension" by Lionni and Rivasseau.

@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation. See, for instance, the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010.

@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation (aka intermediate field representation). See, for instance, the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010. See in particular Sections III.A and IV.A.

@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case if you want to entice people to write useful and thoughtful answer, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking vertices with $p$ legs in a QFT into trivalent vertices, then one can do that by iterating the HB transformation. See for instance the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010.

@HyyFly: It would be nice to elaborate a bit on your question and in particular give some motivation. What applications do you have in mind for possible generalizations of the Hubbard-Stratonovich transformation? The latter is just the physics parlance for the formula for the Fourier transform of a Gaussian, used in reverse. In any case, if you want to entice people to write useful and thoughtful answers, how you write your question must show some effort.

As to the question itself:

The problem of figuring out when "simple functions" like the exponential of a quadratic has a simple Fourier transform was addressed in the article "When is the Fourier transform of an elementary function elementary?" by Etingof, Kazhdan and Polishchuk in Selecta Math. 2002.

If all you want is a way of breaking quantum field theory vertices with $p$ legs into trivalent vertices, then one can do that by iterating the Hubbard-Stratonovich transformation. See, for instance, the article "Loop vertex expansion for $\Phi^{2k}$ theory in zero dimension" by Rivasseau and Wang in J. Math. Phys. 2010.