Convolution in the time domain Multiplication in the frequency domain

Matrix-vector multiplicationConvolutionMultiplication

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Toeplitz Block ToeplitzDoubly infinite Doubly infiniteBanded Banded

Polynomial A(θ) Matrix Polynomial A(θ)

A-1 (θ) = inverse of A(θ) (not banded?)BC (θ) = B(θ)C(θ) (banded)

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Not banded Singly infinite / finite

Periodic A(θ) Circulants

Function theory

Bottcher-Silbermann Szego-Grenander

Grochenig (Wiener L1 Lemma)

Wiener-Hopf Gohberg-Semencul

Factorization A(θ) = U(θ) L(θ)

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A = L U or A = L P U

Plemelj / G.D. Birkhoff / ...Gohberg / Kaashoek / Spitkovsky / ...Can we reach linear factors A = A1 A2 ... Ak?

Banded but not Toeplitz

“Time-varying filter”Symbol A(θ) varies from row to rowFactorization still possible?

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A = L1 P1 U1 A = L1 σ2 L2

A = U1 σ1 U2 A = U2 P2 L2

Problem for a banded doubly infinite matrix: a11 is in the middle of A

Need elimination starting from - ∞ !!

A = L P U is still possible, even if A is not Toeplitz

a11 a1n

an1 ann[ ]

when

/ offset

Blocks in C are offset by w from blocks in B

Which is the main diagonal of a doubly infinite matrix?

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Example: one-way shift S (lower diagonal of 1s)

Inverse = reverse shift ST (upper diagonal of 1s)

Index (S+) = dim (nullspace(S+)) - dim(nullspace((S+)T)) = 0 - 1

Main diagonal of shift matrix S is diagonal -1

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Fredholm: Index not changed if an entry changes

Index(BC) = Index(B) + Index(C)

Index (S+) locates the main diagonal

Index (S+) = 0 when A is centered

Then inverse of upper is upper/ finite sections OK

Banded permutation: Index by counting 1’s (Lindner, GS)

Matrix entries now have 4 indices (not 2)Vector entries now have 2 indices (not 1)

A is a 2D convolution matrixcontains

The symbol is

for this example?when A is banded?

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