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) a , x ∈ R d M a ( f )( x ) := f ( ax ) ax = ( a 1 x 1 , . . . , a d x d )

Hadamard multipliers Multiplicative Fourier-Laplace transform 1. Dilation invariant operators translation invariant operators ↔ convolution oper. ↔ Fourier analysis Definition (dilation) a , x ∈ R d M a ( f )( x ) := f ( ax ) ax = ( a 1 x 1 , . . . , a d x d ) dilation invariant operators ↔ “multiplicative convolution”? ↔ ??

Hadamard multipliers Multiplicative Fourier-Laplace transform 1. Dilation invariant operators translation invariant operators ↔ convolution oper. ↔ Fourier analysis Definition (dilation) a , x ∈ R d M a ( f )( x ) := f ( ax ) ax = ( a 1 x 1 , . . . , a d x d ) 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 ( R d ) — the class of real analytic functions

Hadamard multipliers Multiplicative Fourier-Laplace transform 2. The class of real analytic functions Notation A ( R d ) — 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 ( R d ) — 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 ( R d ) — 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 ( R d ) is continuous iff it is sequentially continuous.

Hadamard multipliers Multiplicative Fourier-Laplace transform 2. The class of real analytic functions Notation A ( R d ) — 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 ( R d ) is continuous iff it is sequentially continuous. Definition f n → f in A ( R d ) iff ∃ U a complex neighbourhood of R d s.t. f n , f ∈ H ( U ) and f n → 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 ( R d ) → A ( R d ) be a linear continuous map. TFAE

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax )

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d ( m α ) α ⊂ C — the multiplier sequence;

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d ( m α ) α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ′ T ∈ A ( R d ) ∃ ! L ( f )( x ) = � T , M x ( f ) �

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d ( m α ) α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ′ T ∈ A ( R d ) ∃ ! L ( f )( x ) = � T , M x ( f ) � = � T , f ( x · ) �

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d ( m α ) α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ′ T ∈ A ( R d ) ∃ ! L ( f )( x ) = � T , M x ( f ) � = � T , f ( x · ) � m α = � T , x α � — the moment sequence.

Hadamard multipliers Multiplicative Fourier-Laplace transform 3. Hadamard multipliers Theorem Let L : A ( R d ) → A ( R d ) be a linear continuous map. TFAE ∀ a ∈ R d ; (a) LM a = M a L M a ( f )( x ) = f ( ax ) Lx α = m α x α , (b) (monomials are eigenvectors) ∀ α ∈ N d ( m α ) α ⊂ C — the multiplier sequence; (c) (multiplicative convolution) ′ T ∈ A ( R d ) ∃ ! L ( f )( x ) = � T , M x ( f ) � = � T , f ( x · ) � m α = � T , x α � — the moment sequence. Definition Operators as above are called (Hadamard) multipliers on A ( R d ) .

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) z ∈ C d — a polynomial P ( z ) = � | α |≤ q a α z α ,

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j | α |≤ q a α θ α P ( θ ) := �

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j | α |≤ q a α θ α = � | α |≤ q a α θ α 1 1 . . . θ α d P ( θ ) := � — an Euler pdo d

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j | α |≤ q a α θ α = � | α |≤ q a α θ α 1 1 . . . θ α d P ( θ ) := � — an Euler pdo d ( P ( α )) α ∈ N d — the multiplier sequence

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j | α |≤ q a α θ α = � | α |≤ q a α θ α 1 1 . . . θ α d P ( θ ) := � — an Euler pdo d ( P ( α )) α ∈ N d — the multiplier sequence Problems

Hadamard multipliers Multiplicative Fourier-Laplace transform 4. Examples Example (Euler partial differential operators) z ∈ C d — a polynomial θ j ( f ) = x j ∂ f P ( z ) = � | α |≤ q a α z α , ∂ x j | α |≤ q a α θ α = � | α |≤ q a α θ α 1 1 . . . θ α d P ( θ ) := � — an Euler pdo d ( P ( α )) α ∈ N d — the multiplier sequence Problems Describe multiplier sequences = moment sequences for analytic functionals.