Hadamard type operators for real analytic functions of several - - PowerPoint PPT Presentation

Hadamard multipliers Multiplicative Fourier-Laplace transform

Hadamard type operators for real analytic functions of several variables and moments of analytic functionals

Paweł Doma´ nski (based on joint results with Michael Langenbruch – Oldenburg)

• A. Mickiewicz University, Pozna´

n, Poland amu.edu.pl/∼domanski

Pełczy´ nski Memorial Conference Be ¸dlewo, 13-19.07.2014

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 1. Dilation invariant operators

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 1. Dilation invariant operators

translation invariant operators ↔ convolution oper. ↔ Fourier analysis

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 1. Dilation invariant operators

translation invariant operators ↔ convolution oper. ↔ Fourier analysis Definition (dilation) Ma(f)(x) := f(ax) ax = (a1x1, . . . , adxd) a, x ∈ Rd

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 1. Dilation invariant operators

translation invariant operators ↔ convolution oper. ↔ Fourier analysis Definition (dilation) Ma(f)(x) := f(ax) ax = (a1x1, . . . , adxd) a, x ∈ Rd dilation invariant operators ↔ “multiplicative convolution”? ↔ ??

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 1. Dilation invariant operators

translation invariant operators ↔ convolution oper. ↔ Fourier analysis Definition (dilation) Ma(f)(x) := f(ax) ax = (a1x1, . . . , adxd) a, x ∈ Rd dilation invariant operators ↔ “multiplicative convolution”? ↔ ?? Difference translations: a group dilations: a semigroup

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Notation A (Rd) — the class of real analytic functions

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Notation A (Rd) — the class of real analytic functions Topology: nice, complete, nuclear, closed graph theorem, uniform boundedness principle

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Notation A (Rd) — the class of real analytic functions Topology: nice, complete, nuclear, closed graph theorem, uniform boundedness principle but non-Banach, non-Fr´ echet...

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Notation A (Rd) — the class of real analytic functions Topology: nice, complete, nuclear, closed graph theorem, uniform boundedness principle but non-Banach, non-Fr´ echet... Fact A linear operator on A (Rd) is continuous iff it is sequentially continuous.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 2. The class of real analytic functions

Notation A (Rd) — the class of real analytic functions Topology: nice, complete, nuclear, closed graph theorem, uniform boundedness principle but non-Banach, non-Fr´ echet... Fact A linear operator on A (Rd) is continuous iff it is sequentially continuous. Definition fn → f in A (Rd) iff ∃ U a complex neighbourhood of Rd s.t. fn, f ∈ H(U) and fn → f in H(U).

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα,

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα, (mα)α ⊂ C — the multiplier sequence;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα, (mα)α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ∃ ! T ∈ A (Rd)

′

L(f)(x) = T, Mx(f)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα, (mα)α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ∃ ! T ∈ A (Rd)

′

L(f)(x) = T, Mx(f) = T, f(x·)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα, (mα)α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ∃ ! T ∈ A (Rd)

′

L(f)(x) = T, Mx(f) = T, f(x·) mα = T, xα — the moment sequence.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 3. Hadamard multipliers

Theorem Let L : A (Rd) → A (Rd) be a linear continuous map. TFAE (a) LMa = MaL ∀ a ∈ Rd; Ma(f)(x) = f(ax) (b) (monomials are eigenvectors) ∀ α ∈ Nd Lxα = mαxα, (mα)α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ∃ ! T ∈ A (Rd)

′

L(f)(x) = T, Mx(f) = T, f(x·) mα = T, xα — the moment sequence. Definition Operators as above are called (Hadamard) multipliers on A (Rd) .

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα = |α|≤q aαθα1 1 . . . θαd d

— an Euler pdo

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα = |α|≤q aαθα1 1 . . . θαd d

— an Euler pdo (P(α))α∈Nd — the multiplier sequence

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα = |α|≤q aαθα1 1 . . . θαd d

— an Euler pdo (P(α))α∈Nd — the multiplier sequence Problems

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα = |α|≤q aαθα1 1 . . . θαd d

— an Euler pdo (P(α))α∈Nd — the multiplier sequence Problems Describe multiplier sequences = moment sequences for analytic functionals.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 4. Examples

Example (Euler partial differential operators) P(z) =

|α|≤q aαzα,

z ∈ Cd — a polynomial θj(f) = xj ∂f

∂xj

P(θ) :=

|α|≤q aαθα = |α|≤q aαθα1 1 . . . θαd d

— an Euler pdo (P(α))α∈Nd — the multiplier sequence Problems Describe multiplier sequences = moment sequences for analytic functionals. Which multipliers can be inverted or “partially inverted”?

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞}

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} T ∈ A (Rd)′

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} T ∈ A (Rd)′: supp T := min. cpct K s.t. T ∈ H(U)′ ∀ U nbhd of K

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} Theorem (Interpolation theorem for multiplier sequences)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} Theorem (Interpolation theorem for multiplier sequences) The map F+ : Ha → A ([0, ea]d)′, F+(f), xα = f(α), α ∈ Nd, is a well-defined linear continuous surjective map

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} Theorem (Interpolation theorem for multiplier sequences) The map F+ : Ha → A ([0, ea]d)′, F+(f), xα = f(α), α ∈ Nd, is a well-defined linear continuous surjective map, where f ∈ ker F+ iff: f(z) = d

k=1 sin(πzk)gk(z)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} Theorem (Interpolation theorem for multiplier sequences) The map F+ : Ha → A ([0, ea]d)′, F+(f), xα = f(α), α ∈ Nd, is a well-defined linear continuous surjective map, where f ∈ ker F+ iff: f(z) = d

k=1 sin(πzk)gk(z), where

supz∈ωd

n |gk(z)| exp

• −
• a + 1

n j=k Re zj + n Re zk

• < ∞.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 5. The main theorem — “Fourier-Laplace transform”

Definition (Class of holomorphic functions) a ∈ R, vn ց −∞, kn ր +∞, ωn = vn + {z = x + iy ∈ C : |y| < knx}. Ha := {f ∈ H(Cd) | f(β) = 0 ∀ β ∈ Zd \ Nd and ∀ n ∈ N sup

z∈ωd

n

|f(z)| exp  −

• a + 1

n

j

Re zj   < ∞} Theorem (Interpolation theorem for multiplier sequences) The map F+ : Ha → A ([0, ea]d)′, F+(f), xα = f(α), α ∈ Nd, is a well-defined linear continuous surjective map, where f ∈ ker F+ iff: f(z) = d

k=1 sin(πzk)gk(z), where

supz∈ωd

n |gk(z)| exp

• −
• a + 1

n j=k Re zj + n Re zk

• < ∞.

For d = 1: Arakelyan 1980, for arbitrary d: D-Langenbruch 2014

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Theorem For any homogeneous polynomial P of d variables TFAE

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd);

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property,

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+: the

product of the right halfplanes.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 6. Surjectivity of Euler operators P(θ)

Notation A+(Rd) := {f ∈ A (Rd) : f =

α>0 fαzα}

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+: the

product of the right halfplanes. Observation im P(θ) ⊃ A+(Rd) iff

• 1

P(α)

• α>0 is a multiplier sequence.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 7. Polynomials with the Hurwitz property

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 7. Polynomials with the Hurwitz property

Theorem (Choe-Oxley-Sokal-Wagner 2004) A homogeneous polynomial with the Hurwitz property is proportional to a polynomial with all coefficients real non-negative.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 7. Polynomials with the Hurwitz property

Theorem (Choe-Oxley-Sokal-Wagner 2004) A homogeneous polynomial with the Hurwitz property is proportional to a polynomial with all coefficients real non-negative. Theorem (Fettweis 1990) Every elementary symmetric polynomial has the Hurwitz property.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 7. Polynomials with the Hurwitz property

Theorem (Choe-Oxley-Sokal-Wagner 2004) A homogeneous polynomial with the Hurwitz property is proportional to a polynomial with all coefficients real non-negative. Theorem (Fettweis 1990) Every elementary symmetric polynomial has the Hurwitz property. Theorem (Fiedler-Gregor 1981) A quadratic form P has the Hurwitz property iff it is proportional to a quadratic form P1 with non-negative real coefficients and with the positive signature n+(P1) = 1.

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?)

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?) θ2

1 + · · · + θ2 k, k > 1: NO

“Laplace-Euler”;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?) θ2

1 + · · · + θ2 k, k > 1: NO

“Laplace-Euler”; θ2

1 − θ2 2 − θ2 3 − · · · − θ2 k, k > 1: NO

“wave-Euler”;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?) θ2

1 + · · · + θ2 k, k > 1: NO

“Laplace-Euler”; θ2

1 − θ2 2 − θ2 3 − · · · − θ2 k, k > 1: NO

“wave-Euler”; θ2

1 + θ2 2 + · · · + θ2 k − θk+1, k > 1: NO

“heat-Euler”;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?) θ2

1 + · · · + θ2 k, k > 1: NO

“Laplace-Euler”; θ2

1 − θ2 2 − θ2 3 − · · · − θ2 k, k > 1: NO

“wave-Euler”; θ2

1 + θ2 2 + · · · + θ2 k − θk+1, k > 1: NO

“heat-Euler”; θ1θ2 + θ2θ3 + θ1θ3: YES;

Hadamard multipliers Multiplicative Fourier-Laplace transform

• 8. Examples of surjective Euler operators

Theorem For any homogeneous polynomial P of d variables TFAE (a) im P(θ) ⊃ A+(Rd); (b) P has the Hurwitz property, i.e., it has no zeros in Cd

+.

A+(Rd) := {f ∈ A (Rd) : f =

α1>0,α2>0,...,αd>0 fαzα}

Example (im P(θ) ⊃ A+(Rd)?) θ2

1 + · · · + θ2 k, k > 1: NO

“Laplace-Euler”; θ2

1 − θ2 2 − θ2 3 − · · · − θ2 k, k > 1: NO

“wave-Euler”; θ2

1 + θ2 2 + · · · + θ2 k − θk+1, k > 1: NO

“heat-Euler”; θ1θ2 + θ2θ3 + θ1θ3: YES; Example (Surjective operator on A (Rd) ) θ1θ2 + θ2θ3 + θ1θ3 + 2(θ1 + θ2 + θ3) + 3