Spectral Synthesis on Affine Groups 6th Workshop on Fourier Analysis - - PowerPoint PPT Presentation

Spectral Synthesis on Affine Groups

6th Workshop on Fourier Analysis and Related Fields 24–31 August, 2017, P´ ecs, Hungary L´ aszl´

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ekelyhidi University of Debrecen – Institute of Mathematics

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ekelyhidi Spectral Synthesis on Affine Groups

The Spectral Synthesis Theorem

Laurent Schwartz, 1947 Spectral synthesis holds on the reals. In other words: given any continuous complex valued function f on the reals it is the uniform limit

• n compact sets of linear combinations of exponential monomials of the

form x ÞÑ xneλx (n is a natural number, λ is a complex number) such that all these exponential monomials belong to the smallest linear space including all translates of f and being closed with respect to uniform convergence on compact sets.

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ekelyhidi Spectral Synthesis on Affine Groups

The Spectral Synthesis Theorem

Laurent Schwartz, 1947 Spectral synthesis holds on the reals. In other words: given any continuous complex valued function f on the reals it is the uniform limit

• n compact sets of linear combinations of exponential monomials of the

form x ÞÑ xneλx (n is a natural number, λ is a complex number) such that all these exponential monomials belong to the smallest linear space including all translates of f and being closed with respect to uniform convergence on compact sets.

Laurent Schwartz With his butterflies

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ekelyhidi Spectral Synthesis on Affine Groups

The Spectral Synthesis Theorem

Laurent Schwartz, 1947 Spectral synthesis holds on the reals. In other words: given any continuous complex valued function f on the reals it is the uniform limit

• n compact sets of linear combinations of exponential monomials of the

form x ÞÑ xneλx (n is a natural number, λ is a complex number) such that all these exponential monomials belong to the smallest linear space including all translates of f and being closed with respect to uniform convergence on compact sets.

Laurent Schwartz With his butterflies

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ekelyhidi Spectral Synthesis on Affine Groups

Counterexamples

No direct extension of Schwartz’s result to Rn is possible: Spectral synthesis fails to hold in Rn for n ě 2 (Dmitrii I. Gurevich, 1975) For each natural number n ě 2 there exist compactly supported measures µ, ν such that the exponential monomial solutions of the system of functional equations µ ˚ f “ 0, ν ˚ f “ 0 do not span a dense subspace in the solution space of this system.

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ekelyhidi Spectral Synthesis on Affine Groups

Counterexamples

No direct extension of Schwartz’s result to Rn is possible: Spectral synthesis fails to hold in Rn for n ě 2 (Dmitrii I. Gurevich, 1975) For each natural number n ě 2 there exist compactly supported measures µ, ν such that the exponential monomial solutions of the system of functional equations µ ˚ f “ 0, ν ˚ f “ 0 do not span a dense subspace in the solution space of this system. Spectral analysis fails to hold in Rn for n ě 2 (Dmitrii I. Gurevich, 1975) For each natural number n ě 2 there exist compactly supported measures µ1, µ2, . . . , µ6 such that the system µk ˚ f “ 0, k “ 1, 2, . . . , 6 has no exponential monomial solution.

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ekelyhidi Spectral Synthesis on Affine Groups

Counterexamples

No direct extension of Schwartz’s result to Rn is possible: Spectral synthesis fails to hold in Rn for n ě 2 (Dmitrii I. Gurevich, 1975) For each natural number n ě 2 there exist compactly supported measures µ, ν such that the exponential monomial solutions of the system of functional equations µ ˚ f “ 0, ν ˚ f “ 0 do not span a dense subspace in the solution space of this system. Spectral analysis fails to hold in Rn for n ě 2 (Dmitrii I. Gurevich, 1975) For each natural number n ě 2 there exist compactly supported measures µ1, µ2, . . . , µ6 such that the system µk ˚ f “ 0, k “ 1, 2, . . . , 6 has no exponential monomial solution.

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G,

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq,

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f Translation:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f Translation: τyf “ δ´y ˚ f

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f Translation: τyf “ δ´y ˚ f Variety, generated variety:

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f Translation: τyf “ δ´y ˚ f Variety, generated variety: closed submodules are exactly the varieties; τpf q

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ekelyhidi Spectral Synthesis on Affine Groups

Notation and terminology

G: locally compact Abelian group, CpGq: locally convex topological vector space of all continuous complex valued functions on G, topology: compact convergence McpGq: measure algebra « the dual of CpGq « linear space of all compactly supported measures G: commutative algebra with identity Convolution: µ ˚ νpf q “ ż ż f px ` yq dµ dν, µ ˚ f pxq “ ż f px ´ yq dµpyq Vector module: CpGq over McpGq Dirac–measure: δypf q “ f pyq, δ0 is the identity in McpGq Convolution operator: f ÞÑ µ ˚ f Translation: τyf “ δ´y ˚ f Variety, generated variety: closed submodules are exactly the varieties; τpf q

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety Spectral analysis:

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety Spectral analysis: there are nonzero finite dimensional subvarieties

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety Spectral analysis: there are nonzero finite dimensional subvarieties Spectral synthesis:

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety Spectral analysis: there are nonzero finite dimensional subvarieties Spectral synthesis: there are sufficiently many finite dimensional subvarieties

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ekelyhidi Spectral Synthesis on Affine Groups

Spectral analysis and synthesis

Spectral analysis for a variety: every nonzero subvariety has a nonzero finite dimensional subvariety ” every nonzero subvariety has a one dimensional subvariety Synthesizable variety: all nonzero finite dimensional subvarieties span a dense subspace Spectral synthesis for a variety: each subvariety is synthesizable Spectral analysis on a group: spectral analysis holds for each variety Spectral synthesis on a group: spectral synthesis holds for each variety Spectral analysis: there are nonzero finite dimensional subvarieties Spectral synthesis: there are sufficiently many finite dimensional subvarieties

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential:

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers:

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.
• 1. f is an exponential.

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.
• 1. f is an exponential.
• 2. τpf q is one dimensional and f p0q “ 1.

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.
• 1. f is an exponential.
• 2. τpf q is one dimensional and f p0q “ 1.
• 3. f is a normalized common eigenfunction of all translation operators

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.
• 1. f is an exponential.
• 2. τpf q is one dimensional and f p0q “ 1.
• 3. f is a normalized common eigenfunction of all translation operators
• 4. f is a normalized common eigenfunction of all convolution operators.

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Basic function classes

Exponential: continuous homomorphism of G into the multiplicative group of nonzero complex numbers: mpx ` yq “ mpxqmpyq, mp0q “ 1 Theorem Let G be a locally compact Abelian group and f : G Ñ C a continuous

• function. Then the following conditions are equivalent.
• 1. f is an exponential.
• 2. τpf q is one dimensional and f p0q “ 1.
• 3. f is a normalized common eigenfunction of all translation operators
• 4. f is a normalized common eigenfunction of all convolution operators.

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Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G.

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G. If f ‰ 0 then m is unique and we say that f corresponds to m and the smallest n with the above property is called the degree of f .

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Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G. If f ‰ 0 then m is unique and we say that f corresponds to m and the smallest n with the above property is called the degree of f . Exponential polynomial:

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Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G. If f ‰ 0 then m is unique and we say that f corresponds to m and the smallest n with the above property is called the degree of f . Exponential polynomial: linear combination of exponential monomials

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ekelyhidi Spectral Synthesis on Affine Groups

Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G. If f ‰ 0 then m is unique and we say that f corresponds to m and the smallest n with the above property is called the degree of f . Exponential polynomial: linear combination of exponential monomials Theorem Let G be an Abelian group. A variety on CG is finite dimensional if and

• nly if it is spanned by exponential monomials.

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Basic function classes

Exponential monomial: We let for each exponential m: ∆m;y “ δ´y ´ mpyqδ0, the modified difference corresponding to m with increment y. Higher

• rder differences:

∆m;y1,y2,...,yn`1 “ Πn`1

k“1∆m;yk.

The continuous function f : G Ñ C is called an exponential monomial if τpf q is finite dimensional and there exists an exponential m and a natural number n such that ∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1, y2, . . . , yn`1 in G. If f ‰ 0 then m is unique and we say that f corresponds to m and the smallest n with the above property is called the degree of f . Exponential polynomial: linear combination of exponential monomials Theorem Let G be an Abelian group. A variety on CG is finite dimensional if and

• nly if it is spanned by exponential monomials.

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Invariant functions and measures

G: locally compact group K: compact subgroup with normalized Haar measure ω K-invariant functions in CpGq: f pkxlq “ f pxq for x in G and k, l in K. These can be identified with the space CpG{{Kq. K-invariant measures in McpKq: for each f in CpGq ż

G

f pxq dµpxq “ ż

G

ż

K

ż

K

f pkxlq dωpkq dωplq dµpxq These can be identified with the functions in the space McpG{{Kq, which can be identified with a closed subalgebra of McpKq.

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ekelyhidi Spectral Synthesis on Affine Groups

Invariant functions and measures

G: locally compact group K: compact subgroup with normalized Haar measure ω K-invariant functions in CpGq: f pkxlq “ f pxq for x in G and k, l in K. These can be identified with the space CpG{{Kq. K-invariant measures in McpKq: for each f in CpGq ż

G

f pxq dµpxq “ ż

G

ż

K

ż

K

f pkxlq dωpkq dωplq dµpxq These can be identified with the functions in the space McpG{{Kq, which can be identified with a closed subalgebra of McpKq. The pair pG, Kq is called a Gelfand pair if the algebra McpG{{Kq is commutative.

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ekelyhidi Spectral Synthesis on Affine Groups

Invariant functions and measures

G: locally compact group K: compact subgroup with normalized Haar measure ω K-invariant functions in CpGq: f pkxlq “ f pxq for x in G and k, l in K. These can be identified with the space CpG{{Kq. K-invariant measures in McpKq: for each f in CpGq ż

G

f pxq dµpxq “ ż

G

ż

K

ż

K

f pkxlq dωpkq dωplq dµpxq These can be identified with the functions in the space McpG{{Kq, which can be identified with a closed subalgebra of McpKq. The pair pG, Kq is called a Gelfand pair if the algebra McpG{{Kq is commutative. The dual of CpG{{Kq can be identified with McpG{{Kq.

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ekelyhidi Spectral Synthesis on Affine Groups

Invariant functions and measures

G: locally compact group K: compact subgroup with normalized Haar measure ω K-invariant functions in CpGq: f pkxlq “ f pxq for x in G and k, l in K. These can be identified with the space CpG{{Kq. K-invariant measures in McpKq: for each f in CpGq ż

G

f pxq dµpxq “ ż

G

ż

K

ż

K

f pkxlq dωpkq dωplq dµpxq These can be identified with the functions in the space McpG{{Kq, which can be identified with a closed subalgebra of McpKq. The pair pG, Kq is called a Gelfand pair if the algebra McpG{{Kq is commutative. The dual of CpG{{Kq can be identified with McpG{{Kq.

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Projection

The projection f ÞÑ f # on CpGq is defined as f #pxq “ ż

K

ż

K

f pkxlq dωpkq dωplq for x P G. The projection µ ÞÑ µ# on McpGq is defined as xµ#, f y “ ż

G

f #pxq dµpxq for f P CpGq. Then f ÞÑ f # is a continuous linear mapping from CpGq onto CpG{{Kq and its adjoint is µ ÞÑ µ#: xµ, f #y “ xµ#, f y further f is K-invariant if and only if f “ f # and µ is K-invariant if and

• nly if µ “ µ#.

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Projection

The projection f ÞÑ f # on CpGq is defined as f #pxq “ ż

K

ż

K

f pkxlq dωpkq dωplq for x P G. The projection µ ÞÑ µ# on McpGq is defined as xµ#, f y “ ż

G

f #pxq dµpxq for f P CpGq. Then f ÞÑ f # is a continuous linear mapping from CpGq onto CpG{{Kq and its adjoint is µ ÞÑ µ#: xµ, f #y “ xµ#, f y further f is K-invariant if and only if f “ f # and µ is K-invariant if and

• nly if µ “ µ#.

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K-translations and K-varieties

Suppose that pG, Kq is a Gelfand pair. Then the measures δ#

y commute

for all y in G: for each f in CpG{{Kq we have xδ#

y ˚ δ# z , f y “

ż

K

f pykzq dωpkq “ ż

K

f pzkyq dωpkq. Similarly, the operators τy defined on CpG{{Kq by τyf “ δ#

y ´1 ˚ f “

ż f pxz´1q dδ#

y ´1pzq “

ż

K

f pxkyq dωpkq form a commuting family for y in G: these are the K-translations. K-translation invariant closed linear subspaces of CpGq are called K-varieties.

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K-translations and K-varieties

Suppose that pG, Kq is a Gelfand pair. Then the measures δ#

y commute

for all y in G: for each f in CpG{{Kq we have xδ#

y ˚ δ# z , f y “

ż

K

f pykzq dωpkq “ ż

K

f pzkyq dωpkq. Similarly, the operators τy defined on CpG{{Kq by τyf “ δ#

y ´1 ˚ f “

ż f pxz´1q dδ#

y ´1pzq “

ż

K

f pxkyq dωpkq form a commuting family for y in G: these are the K-translations. K-translation invariant closed linear subspaces of CpGq are called K-varieties. One-dimensional K-varieties are spanned by K-spherical functions which are the common K-invariant eigenfunctions s of all K-translations: τys “ s for each y in G: ż

K

f pxkyq dωpkq “ f pxqf pyq.

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K-translations and K-varieties

Suppose that pG, Kq is a Gelfand pair. Then the measures δ#

y commute

for all y in G: for each f in CpG{{Kq we have xδ#

y ˚ δ# z , f y “

ż

K

f pykzq dωpkq “ ż

K

f pzkyq dωpkq. Similarly, the operators τy defined on CpG{{Kq by τyf “ δ#

y ´1 ˚ f “

ż f pxz´1q dδ#

y ´1pzq “

ż

K

f pxkyq dωpkq form a commuting family for y in G: these are the K-translations. K-translation invariant closed linear subspaces of CpGq are called K-varieties. One-dimensional K-varieties are spanned by K-spherical functions which are the common K-invariant eigenfunctions s of all K-translations: τys “ s for each y in G: ż

K

f pxkyq dωpkq “ f pxqf pyq.

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K-spectral analysis

We say that K-spectral analysis holds for a K-variety if every nonzero K-subvariety of it contains a K-spherical function. We say that K-spectral analysis holds for G if K-spectral analysis holds for every K-variety. In a commutative complex algebra A a maximal ideal M is called exponential maximal ideal, if A{M is isomorphic to the complex filed. K-spectral analysis K-spectral analysis holds for the K-variety V if and only if for every closed maximal ideal M of the residue algebra McpG{{Kq M Ann V is

• exponential. K-spectral analysis holds for G if and only if every closed

maximal ideal of McpG{{Kq is exponential.

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K-spectral analysis

We say that K-spectral analysis holds for a K-variety if every nonzero K-subvariety of it contains a K-spherical function. We say that K-spectral analysis holds for G if K-spectral analysis holds for every K-variety. In a commutative complex algebra A a maximal ideal M is called exponential maximal ideal, if A{M is isomorphic to the complex filed. K-spectral analysis K-spectral analysis holds for the K-variety V if and only if for every closed maximal ideal M of the residue algebra McpG{{Kq M Ann V is

• exponential. K-spectral analysis holds for G if and only if every closed

maximal ideal of McpG{{Kq is exponential.

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Modified differences and K-monomials

For each K-spherical function s we define the modified K-difference ∆s;y “ δ#

y ´1 ´ spyqδe,

and their products ∆s;y1,y2,...,yk`1 “ Πk`1

j“1 ∆s;yj. Given the K-spherical

function s the closed ideal generated by all modified differences ∆s;y with y in G is an exponential maximal ideal, denoted by Ms. The K-invariant f is called an s-monomial if dim τKpf q ă 8 is and there is a natural number k such that Mk`1

s

Ď Ann τKpf q where τKpf q denotes the K-variety generated by f . This is equivalent to the functional equation ∆s;y1,y2,...,yk`1 ˚ f pxq “ 0 for each x, y1, y2, . . . , yk`1 in G. If f is nonzero, then s is uniquely determined, and the smallest k with this property is called the degree of the s-monomial f .

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K-spectral synthesis

For instance, s-monomials of degree 2 are of the form cs ` f , where f is a K-invariant continuous solutions of the K-sine equation: ż

K

f pxkyq dωpkq “ f pxqspyq ` f pyqspxq.

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K-spectral synthesis

For instance, s-monomials of degree 2 are of the form cs ` f , where f is a K-invariant continuous solutions of the K-sine equation: ż

K

f pxkyq dωpkq “ f pxqspyq ` f pyqspxq. We say that the K-variety is K-synthesizable if all K-monomials span a dense subspace in the variety. We say that K-spectral synthesis holds for a K-variety, if every nonzero subvariety of it is K-synthesizable. K-spectral synthesis implies K-spectral analysis.

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K-spectral synthesis

For instance, s-monomials of degree 2 are of the form cs ` f , where f is a K-invariant continuous solutions of the K-sine equation: ż

K

f pxkyq dωpkq “ f pxqspyq ` f pyqspxq. We say that the K-variety is K-synthesizable if all K-monomials span a dense subspace in the variety. We say that K-spectral synthesis holds for a K-variety, if every nonzero subvariety of it is K-synthesizable. K-spectral synthesis implies K-spectral analysis. K-synthesizability The K-variety V is synthesizable if and only if its annihilator is the intersection of those cofinite closed ideals of McpG{{Kq which contain it.

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K-spectral synthesis

For instance, s-monomials of degree 2 are of the form cs ` f , where f is a K-invariant continuous solutions of the K-sine equation: ż

K

f pxkyq dωpkq “ f pxqspyq ` f pyqspxq. We say that the K-variety is K-synthesizable if all K-monomials span a dense subspace in the variety. We say that K-spectral synthesis holds for a K-variety, if every nonzero subvariety of it is K-synthesizable. K-spectral synthesis implies K-spectral analysis. K-synthesizability The K-variety V is synthesizable if and only if its annihilator is the intersection of those cofinite closed ideals of McpG{{Kq which contain it. If K is a normal subgroup and G{K is commutative, then all these concepts coincide with the corresponding spectral analysis and synthesis concepts on the locally compact Abelian group G{K. Obviously, this is the case if G itself is commutative.

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K-spectral synthesis

For instance, s-monomials of degree 2 are of the form cs ` f , where f is a K-invariant continuous solutions of the K-sine equation: ż

K

f pxkyq dωpkq “ f pxqspyq ` f pyqspxq. We say that the K-variety is K-synthesizable if all K-monomials span a dense subspace in the variety. We say that K-spectral synthesis holds for a K-variety, if every nonzero subvariety of it is K-synthesizable. K-spectral synthesis implies K-spectral analysis. K-synthesizability The K-variety V is synthesizable if and only if its annihilator is the intersection of those cofinite closed ideals of McpG{{Kq which contain it. If K is a normal subgroup and G{K is commutative, then all these concepts coincide with the corresponding spectral analysis and synthesis concepts on the locally compact Abelian group G{K. Obviously, this is the case if G itself is commutative.

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Semidirect products

Let N be a locally compact topological group and K is a compact group

• f automorphisms of N. We consider the semidirect product of K and N:

K ˙ N: it is K ˆ N equipped with the operation pk, nq ¨ pl, mq “ pk ˝ l, pk ¨ mqnq, where ˝ is the composition of the automorphisms k, l, ¨ is the effect of the automorphisms on the elements of N, and juxtaposition is the group

• peration in N. It turns out that this operation defines a group structure
• n K ˆ N, where the identity is pid, eq, with the identity automorphism

id of N and the identity element e of n, and the inverse of pk, nq is pk´1, k´1 ¨ u´1q. With the product topology G “ K ˙ N is a locally compact topological group, the semidirect product of K and N. The group N is topologically isomorphic to the closed normal subgroup tpid, nq : n P Nu, and the group K is topologically isomorphic to the compact subgroup tpk, eq : k P Ku. We shall identify these isomorphic groups: K “ tpk, eq : k P Ku, N “ tpid, nq : n P Nu.

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Semidirect products

Let N be a locally compact topological group and K is a compact group

• f automorphisms of N. We consider the semidirect product of K and N:

K ˙ N: it is K ˆ N equipped with the operation pk, nq ¨ pl, mq “ pk ˝ l, pk ¨ mqnq, where ˝ is the composition of the automorphisms k, l, ¨ is the effect of the automorphisms on the elements of N, and juxtaposition is the group

• peration in N. It turns out that this operation defines a group structure
• n K ˆ N, where the identity is pid, eq, with the identity automorphism

id of N and the identity element e of n, and the inverse of pk, nq is pk´1, k´1 ¨ u´1q. With the product topology G “ K ˙ N is a locally compact topological group, the semidirect product of K and N. The group N is topologically isomorphic to the closed normal subgroup tpid, nq : n P Nu, and the group K is topologically isomorphic to the compact subgroup tpk, eq : k P Ku. We shall identify these isomorphic groups: K “ tpk, eq : k P Ku, N “ tpid, nq : n P Nu.

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Example: Affine groups

Let X be a finite dimensional vector space and K a compact subgroup of GLpXq, the general linear group of X with the normalized Haar measure ω. Then the set K ˆ X acts on X: for S in K and u in V let pS, uqx defined by the affine mapping pS, uqx “ Sx ` u for each x in V . The composition of affine mappings defines the

• peration on K ˆ X as

pS, uq ¨ pT, vq “ pS ˝ T, Sv ` uq and with the identity pid, 0q and inverse pS, uq´1 “ pS´1, ´S´1uq we

• btain the group

Aff K “ K ˙ X, the semidirect product of K and X. Here – as we have seen – K is topologically isomorphic to the compact subgroup tpS, 0q : S P Ku and X is topologically isomorphic to the closed normal subgroup tpid, uq : u P Xu.

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Example: Semidirect products

K-invariant functions are exactly those functions pS, uq ÞÑ f pS, uq which depend only on u and are invariant with respect to K: f pS, uq “ f pid, uq “ f pid, Suq for each S in K and u in X. Hence CpAff K{{Kq can be identified with a closed subspace of CpXq, the space of K-radial functions. Similarly, the space of K-invariant measures McpAff K{{Kq on Aff K can be identified with a closed subspace of McpXq, the space of K-radial measures, Then Aff K is a locally compact group, K is topologically isomorphic to the compact subgroup tpL, 0q : L P Ku, and Rn is topologically isomorphic to the normal subgroup tpid, uq : u P Rnu.

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Example: Semidirect products

K-invariant functions are exactly those functions pS, uq ÞÑ f pS, uq which depend only on u and are invariant with respect to K: f pS, uq “ f pid, uq “ f pid, Suq for each S in K and u in X. Hence CpAff K{{Kq can be identified with a closed subspace of CpXq, the space of K-radial functions. Similarly, the space of K-invariant measures McpAff K{{Kq on Aff K can be identified with a closed subspace of McpXq, the space of K-radial measures, Then Aff K is a locally compact group, K is topologically isomorphic to the compact subgroup tpL, 0q : L P Ku, and Rn is topologically isomorphic to the normal subgroup tpid, uq : u P Rnu.

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Example: The Poincar´ e group

The Poincar´ e group We consider the real vector space R1,3 “ R ‘ R3 equipped with the indefinite inner product xv, wy “ v0w0 ´

3

ÿ

j“1

vjwj, where v “ pv0, v1, v2, v3q and w “ pw0, w1, w2, w3q. The isometry group Op1, 3q of this indefinite inner product space is called the Lorentz group. The affine group of the Lorentz group Aff Op1, 3q “ Op1, 3q ˙ R1,3 is the Poincar´ e group.

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Example: The Poincar´ e group

The Poincar´ e group We consider the real vector space R1,3 “ R ‘ R3 equipped with the indefinite inner product xv, wy “ v0w0 ´

3

ÿ

j“1

vjwj, where v “ pv0, v1, v2, v3q and w “ pw0, w1, w2, w3q. The isometry group Op1, 3q of this indefinite inner product space is called the Lorentz group. The affine group of the Lorentz group Aff Op1, 3q “ Op1, 3q ˙ R1,3 is the Poincar´ e group.

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Example: Euclidean motions

The group of Euclidean motions We consider the vector space Rn and the orthogonal group Opnq, the group of rotations. Together with translations they generate the group of Euclidean motions: rigid motions leaving the origin fixed. This is the affine group of Opnq: Aff Opnq “ Opnq ˙ Rn which acts on Rn by pO, uqx “ Ox ` u for O in Opnq and x, u in Rn. Clearly, for n “ 1 we have Op1q “ t`1, ´1u. Op1q-spherical functions are the functions of the form x ÞÑ cosh λx with arbitrary complex λ.

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Example: Euclidean motions

The group of Euclidean motions We consider the vector space Rn and the orthogonal group Opnq, the group of rotations. Together with translations they generate the group of Euclidean motions: rigid motions leaving the origin fixed. This is the affine group of Opnq: Aff Opnq “ Opnq ˙ Rn which acts on Rn by pO, uqx “ Ox ` u for O in Opnq and x, u in Rn. Clearly, for n “ 1 we have Op1q “ t`1, ´1u. Op1q-spherical functions are the functions of the form x ÞÑ cosh λx with arbitrary complex λ.

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Example: Proper Euclidean motions

The group of proper motions We consider the vector space Rn and the special orthogonal group SOpnq, the group of proper rotations: orthogonal operators with determinant `1. Together with translations they generate the group of proper Euclidean motions: rigid motions which preserve orientation: no reflection is included. This is the affine group of SOpnq: Aff SOpnq “ SOpnq ˙ Rn which acts on Rn by pS, uqx “ Sx ` u for S in SOpnq and x, u in Rn. Clearly, for n “ 1 we have SOp1q “ tidu, hence Aff SOp1q “ R – in one dimension the proper Euclidean motions are exactly the translations.

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Example: Proper Euclidean motions

The group of proper motions We consider the vector space Rn and the special orthogonal group SOpnq, the group of proper rotations: orthogonal operators with determinant `1. Together with translations they generate the group of proper Euclidean motions: rigid motions which preserve orientation: no reflection is included. This is the affine group of SOpnq: Aff SOpnq “ SOpnq ˙ Rn which acts on Rn by pS, uqx “ Sx ` u for S in SOpnq and x, u in Rn. Clearly, for n “ 1 we have SOp1q “ tidu, hence Aff SOp1q “ R – in one dimension the proper Euclidean motions are exactly the translations.

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Example: Proper Euclidean motions

The SOpnq-invariant functions can be identified with those continuous functions f : Rn Ñ C with f pSxq “ f pxq for each S in SOpnq and x in

• Rn. These are called radial functions as f pxq depends only on }x}:

f pxq “ ϕp}x}q for some continuous ϕ : R Ñ C. Similarly, McpAff SOpnqq is identified with those measures in McpRnq with ż

Rn f pSxq dµpxq “

ż

Rn f pxq dµpxq

for each f in CpRnq and S in SOpnq: radial measures.

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Example: Proper Euclidean motions

The SOpnq-invariant functions can be identified with those continuous functions f : Rn Ñ C with f pSxq “ f pxq for each S in SOpnq and x in

• Rn. These are called radial functions as f pxq depends only on }x}:

f pxq “ ϕp}x}q for some continuous ϕ : R Ñ C. Similarly, McpAff SOpnqq is identified with those measures in McpRnq with ż

Rn f pSxq dµpxq “

ż

Rn f pxq dµpxq

for each f in CpRnq and S in SOpnq: radial measures. Convolution in McpAff SOpnq{{SOpnqq coincides with the ordinary convolution in Rn, hence pAff SOpnq, SOpnqq is a Gelfand pair. Radial functions: CrpRnq « CpAff pSOpnqq{{SOpnqq Radial measures: MrpRnq « McpAff pSOpnqq{{SOpnqq

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Example: Proper Euclidean motions

The SOpnq-invariant functions can be identified with those continuous functions f : Rn Ñ C with f pSxq “ f pxq for each S in SOpnq and x in

• Rn. These are called radial functions as f pxq depends only on }x}:

f pxq “ ϕp}x}q for some continuous ϕ : R Ñ C. Similarly, McpAff SOpnqq is identified with those measures in McpRnq with ż

Rn f pSxq dµpxq “

ż

Rn f pxq dµpxq

for each f in CpRnq and S in SOpnq: radial measures. Convolution in McpAff SOpnq{{SOpnqq coincides with the ordinary convolution in Rn, hence pAff SOpnq, SOpnqq is a Gelfand pair. Radial functions: CrpRnq « CpAff pSOpnqq{{SOpnqq Radial measures: MrpRnq « McpAff pSOpnqq{{SOpnqq

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Example: Proper Euclidean motions

SOpnq-translation: for f in CrpRnq and y in Rn τypf qpxq “ ż

SOpnq

f px ` kyq dωpkq SOpnq-variety: V Ď CrpRnq linear subspace, closed with respect to uniform convergence on compact sets, and for each f in V and y in Rn we have x ÞÑ ş

SOpnq f px ` kyq dωpkq is in V

SOpnq-spherical function: s ‰ 0 in CrpRnq and ż

SOpnq

spx ` kyq dωpkq “ spxqspyq for each y P Rn

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Example: Proper Euclidean motions

Eigenfunctions of the Laplacian The SOpnq-spherical functions are exactly the normalized radial eigenfunctions of the Laplacian in Rn. Let ϕp}x}q “ spxq for x P Rn, then, using the radial form of the Laplacian in Rn we have the Bessel differential equation d2 dr 2 ϕprq ` n ´ 1 r d dr ϕprq “ λϕprq, with ϕ is regular at 0 and ϕp0q “ 1. Let Jλ denote the function Jλprq “ Γ `n 2 ˘

8

ÿ

k“0

λk k! Γpk ` n

2q

´r 2 ¯2k . Then s is an SOpnq-spherical function if and only if spxq “ sλpxq “ Jλp}x}q holds for each x in Rn with some complex number λ.

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Example: Proper Euclidean motions

Eigenfunctions of the Laplacian The SOpnq-spherical functions are exactly the normalized radial eigenfunctions of the Laplacian in Rn. Let ϕp}x}q “ spxq for x P Rn, then, using the radial form of the Laplacian in Rn we have the Bessel differential equation d2 dr 2 ϕprq ` n ´ 1 r d dr ϕprq “ λϕprq, with ϕ is regular at 0 and ϕp0q “ 1. Let Jλ denote the function Jλprq “ Γ `n 2 ˘

8

ÿ

k“0

λk k! Γpk ` n

2q

´r 2 ¯2k . Then s is an SOpnq-spherical function if and only if spxq “ sλpxq “ Jλp}x}q holds for each x in Rn with some complex number λ.

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SOpnq-monomials

Derivatives with respect to the parameter Given the SOpnq-spherical function sλ with some complex λ the sλ-monomials of degree at most k are exactly the linear combinations of the derivatives

dj dλj sλ for j “ 0, 1, . . . , k.

SOpnq-spectral analysis and synthesis Every nonzero variety contains an SOpnq-spherical function, moreover, all functions of the form

dj dλj sλ span a dense subspace in every variety.

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SOpnq-monomials

Derivatives with respect to the parameter Given the SOpnq-spherical function sλ with some complex λ the sλ-monomials of degree at most k are exactly the linear combinations of the derivatives

dj dλj sλ for j “ 0, 1, . . . , k.

SOpnq-spectral analysis and synthesis Every nonzero variety contains an SOpnq-spherical function, moreover, all functions of the form

dj dλj sλ span a dense subspace in every variety.

As SOp1q “ tidu, hence Aff SOp1q “ R, SOp1q-varieties are exactly the closed translation invariant subspaces of CpRq. SOp1q-spherical functions are exactly the exponentials: sλpxq “ eλx, and SOp1q-monomials are the linear combinations of the functions dj dλj sλpxq “ xjeλx. Our spectral synthesis theorem is a proper generalization of L. Schwartz’s theorem to Rn.

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SOpnq-monomials

Derivatives with respect to the parameter Given the SOpnq-spherical function sλ with some complex λ the sλ-monomials of degree at most k are exactly the linear combinations of the derivatives

dj dλj sλ for j “ 0, 1, . . . , k.

SOpnq-spectral analysis and synthesis Every nonzero variety contains an SOpnq-spherical function, moreover, all functions of the form

dj dλj sλ span a dense subspace in every variety.

As SOp1q “ tidu, hence Aff SOp1q “ R, SOp1q-varieties are exactly the closed translation invariant subspaces of CpRq. SOp1q-spherical functions are exactly the exponentials: sλpxq “ eλx, and SOp1q-monomials are the linear combinations of the functions dj dλj sλpxq “ xjeλx. Our spectral synthesis theorem is a proper generalization of L. Schwartz’s theorem to Rn.

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