Topological properties of convolutor spaces via the short-time - PowerPoint PPT Presentation

Topological properties of convolutor spaces via the short-time Fourier transform Andreas Debrouwere (Joint work with Jasson Vindas) Ghent University Pawel Doma´ nski Memorial Conference 6 July 2018 1 / 14

The space of integrable distributions (1) The space B consists of all ϕ ∈ C ∞ ( R d ) such that � ∂ α ϕ � L ∞ < ∞ , ∀ α ∈ N d . The space B is a Fr´ echet space. The space ˙ B is given by the closure of D ( R d ) in B , i.e. it consists of all ϕ ∈ C ∞ ( R d ) such that | x |→∞ ∂ α ϕ ( x ) = 0 , ∀ α ∈ N d . lim The space ˙ B is a Fr´ echet space. The space D ′ L 1 of integrable distributions is given by the topological dual of ˙ B . 2 / 14

The space of integrable distributions (1) The space B consists of all ϕ ∈ C ∞ ( R d ) such that � ∂ α ϕ � L ∞ < ∞ , ∀ α ∈ N d . The space B is a Fr´ echet space. The space ˙ B is given by the closure of D ( R d ) in B , i.e. it consists of all ϕ ∈ C ∞ ( R d ) such that | x |→∞ ∂ α ϕ ( x ) = 0 , ∀ α ∈ N d . lim The space ˙ B is a Fr´ echet space. The space D ′ L 1 of integrable distributions is given by the topological dual of ˙ B . 2 / 14

The space of integrable distributions (1) The space B consists of all ϕ ∈ C ∞ ( R d ) such that � ∂ α ϕ � L ∞ < ∞ , ∀ α ∈ N d . The space B is a Fr´ echet space. The space ˙ B is given by the closure of D ( R d ) in B , i.e. it consists of all ϕ ∈ C ∞ ( R d ) such that | x |→∞ ∂ α ϕ ( x ) = 0 , ∀ α ∈ N d . lim The space ˙ B is a Fr´ echet space. The space D ′ L 1 of integrable distributions is given by the topological dual of ˙ B . 2 / 14

The space of integrable distributions (1) The space B consists of all ϕ ∈ C ∞ ( R d ) such that � ∂ α ϕ � L ∞ < ∞ , ∀ α ∈ N d . The space B is a Fr´ echet space. The space ˙ B is given by the closure of D ( R d ) in B , i.e. it consists of all ϕ ∈ C ∞ ( R d ) such that | x |→∞ ∂ α ϕ ( x ) = 0 , ∀ α ∈ N d . lim The space ˙ B is a Fr´ echet space. The space D ′ L 1 of integrable distributions is given by the topological dual of ˙ B . 2 / 14

The space of integrable distributions (1) The space B consists of all ϕ ∈ C ∞ ( R d ) such that � ∂ α ϕ � L ∞ < ∞ , ∀ α ∈ N d . The space B is a Fr´ echet space. The space ˙ B is given by the closure of D ( R d ) in B , i.e. it consists of all ϕ ∈ C ∞ ( R d ) such that | x |→∞ ∂ α ϕ ( x ) = 0 , ∀ α ∈ N d . lim The space ˙ B is a Fr´ echet space. The space D ′ L 1 of integrable distributions is given by the topological dual of ˙ B . 2 / 14

The space of integrable distributions (2) Theorem (Schwartz, 1950) L 1 if and only if f ∗ ϕ ∈ L 1 for all ϕ ∈ D ( R d ) . Let f ∈ D ′ ( R d ) . Then, f ∈ D ′ Two natural topologies on D ′ L 1 : L 1 , ˙ The strong topology b ( D ′ B ). 1 The initial topology op w.r.t. the mapping 2 L 1 → L b ( D ( R d ) , L 1 ) : f → ( ϕ → f ∗ ϕ ) . D ′ Theorem (Schwartz, 1950) The spaces D ′ L 1 , b and D ′ L 1 , op have the same bounded sets and null sequences. Do the topologies b and op coincide on D ′ L 1 ? 3 / 14

The space of integrable distributions (2) Theorem (Schwartz, 1950) L 1 if and only if f ∗ ϕ ∈ L 1 for all ϕ ∈ D ( R d ) . Let f ∈ D ′ ( R d ) . Then, f ∈ D ′ Two natural topologies on D ′ L 1 : L 1 , ˙ The strong topology b ( D ′ B ). 1 The initial topology op w.r.t. the mapping 2 L 1 → L b ( D ( R d ) , L 1 ) : f → ( ϕ → f ∗ ϕ ) . D ′ Theorem (Schwartz, 1950) The spaces D ′ L 1 , b and D ′ L 1 , op have the same bounded sets and null sequences. Do the topologies b and op coincide on D ′ L 1 ? 3 / 14

The space of integrable distributions (2) Theorem (Schwartz, 1950) L 1 if and only if f ∗ ϕ ∈ L 1 for all ϕ ∈ D ( R d ) . Let f ∈ D ′ ( R d ) . Then, f ∈ D ′ Two natural topologies on D ′ L 1 : L 1 , ˙ The strong topology b ( D ′ B ). 1 The initial topology op w.r.t. the mapping 2 L 1 → L b ( D ( R d ) , L 1 ) : f → ( ϕ → f ∗ ϕ ) . D ′ Theorem (Schwartz, 1950) The spaces D ′ L 1 , b and D ′ L 1 , op have the same bounded sets and null sequences. Do the topologies b and op coincide on D ′ L 1 ? 3 / 14

The space of integrable distributions (2) Theorem (Schwartz, 1950) L 1 if and only if f ∗ ϕ ∈ L 1 for all ϕ ∈ D ( R d ) . Let f ∈ D ′ ( R d ) . Then, f ∈ D ′ Two natural topologies on D ′ L 1 : L 1 , ˙ The strong topology b ( D ′ B ). 1 The initial topology op w.r.t. the mapping 2 L 1 → L b ( D ( R d ) , L 1 ) : f → ( ϕ → f ∗ ϕ ) . D ′ Theorem (Schwartz, 1950) The spaces D ′ L 1 , b and D ′ L 1 , op have the same bounded sets and null sequences. Do the topologies b and op coincide on D ′ L 1 ? 3 / 14

The space of rapidly decreasing distributions (1) The space O C consists of all ϕ ∈ C ∞ ( R d ) such that there is N ∈ N for which | ∂ α ϕ ( x ) | ∀ α ∈ N d . sup (1 + | x | ) N < ∞ , x ∈ R d O C is an ( LF )-space (countable inductive limit of Fr´ echet spaces). The space O ′ C of rapidly decreasing distributions is given by the topological dual of O C . 4 / 14

The space of rapidly decreasing distributions (1) The space O C consists of all ϕ ∈ C ∞ ( R d ) such that there is N ∈ N for which | ∂ α ϕ ( x ) | ∀ α ∈ N d . sup (1 + | x | ) N < ∞ , x ∈ R d O C is an ( LF )-space (countable inductive limit of Fr´ echet spaces). The space O ′ C of rapidly decreasing distributions is given by the topological dual of O C . 4 / 14

The space of rapidly decreasing distributions (2) Theorem (Schwartz, 1950) Let f ∈ S ′ ( R d ) . Then, f ∈ O ′ C if and only if f ∗ ϕ ∈ S ( R d ) for all ϕ ∈ S ( R d ) . O ′ C is sometimes called the space of convolutors of S ( R d ). Define the topologies b and op on O ′ C as before. Theorem (Grothendieck, 1955) The space O ′ C , op is complete, semi-reflexive, and bornological. Consequently, O ′ C , b = O ′ C , op and the ( LF ) -space O C is complete. He showed that O ′ C , op is isomorphic to a complemented subspace of ⊗ s ′ and proved that s � ⊗ s ′ is bornological. Moreover, he showed that s � ( O ′ C , op ) ′ b = O C . 5 / 14

The space of rapidly decreasing distributions (2) Theorem (Schwartz, 1950) Let f ∈ S ′ ( R d ) . Then, f ∈ O ′ C if and only if f ∗ ϕ ∈ S ( R d ) for all ϕ ∈ S ( R d ) . O ′ C is sometimes called the space of convolutors of S ( R d ). Define the topologies b and op on O ′ C as before. Theorem (Grothendieck, 1955) The space O ′ C , op is complete, semi-reflexive, and bornological. Consequently, O ′ C , b = O ′ C , op and the ( LF ) -space O C is complete. He showed that O ′ C , op is isomorphic to a complemented subspace of ⊗ s ′ and proved that s � ⊗ s ′ is bornological. Moreover, he showed that s � ( O ′ C , op ) ′ b = O C . 5 / 14

The space of rapidly decreasing distributions (2) Theorem (Schwartz, 1950) Let f ∈ S ′ ( R d ) . Then, f ∈ O ′ C if and only if f ∗ ϕ ∈ S ( R d ) for all ϕ ∈ S ( R d ) . O ′ C is sometimes called the space of convolutors of S ( R d ). Define the topologies b and op on O ′ C as before. Theorem (Grothendieck, 1955) The space O ′ C , op is complete, semi-reflexive, and bornological. Consequently, O ′ C , b = O ′ C , op and the ( LF ) -space O C is complete. He showed that O ′ C , op is isomorphic to a complemented subspace of ⊗ s ′ and proved that s � ⊗ s ′ is bornological. Moreover, he showed that s � ( O ′ C , op ) ′ b = O C . 5 / 14

The space of rapidly decreasing distributions (2) Theorem (Schwartz, 1950) Let f ∈ S ′ ( R d ) . Then, f ∈ O ′ C if and only if f ∗ ϕ ∈ S ( R d ) for all ϕ ∈ S ( R d ) . O ′ C is sometimes called the space of convolutors of S ( R d ). Define the topologies b and op on O ′ C as before. Theorem (Grothendieck, 1955) The space O ′ C , op is complete, semi-reflexive, and bornological. Consequently, O ′ C , b = O ′ C , op and the ( LF ) -space O C is complete. He showed that O ′ C , op is isomorphic to a complemented subspace of ⊗ s ′ and proved that s � ⊗ s ′ is bornological. Moreover, he showed that s � ( O ′ C , op ) ′ b = O C . 5 / 14

The space of rapidly decreasing distributions (2) Theorem (Schwartz, 1950) Let f ∈ S ′ ( R d ) . Then, f ∈ O ′ C if and only if f ∗ ϕ ∈ S ( R d ) for all ϕ ∈ S ( R d ) . O ′ C is sometimes called the space of convolutors of S ( R d ). Define the topologies b and op on O ′ C as before. Theorem (Grothendieck, 1955) The space O ′ C , op is complete, semi-reflexive, and bornological. Consequently, O ′ C , b = O ′ C , op and the ( LF ) -space O C is complete. He showed that O ′ C , op is isomorphic to a complemented subspace of ⊗ s ′ and proved that s � ⊗ s ′ is bornological. Moreover, he showed that s � ( O ′ C , op ) ′ b = O C . 5 / 14