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• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distn.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/2012-13_notes/04_blevi_sobolev.pdf

• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distn.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/03b_intro_blevi.pdf

• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distns.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/2012-13_notes/04_blevi_sobolev.pdf

• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distn.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/2012-13_notes/04_blevi_sobolev.pdf

EDIT: inserted some links...

• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distns.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter.

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/2012-13_notes/04_blevi_sobolev.pdf

EDIT: inserted some links... EDIT-EDIT: one more...

• Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

• Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

• (Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distns.pdf

• Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

• Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

• Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf (This presumes Poisson summation for $\mathbb A/k$...)

• Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

• The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series always_converge... if only in a suitable Sobolev space. http://www.math.umn.edu/~garrett/m/fun/2012-13_notes/04_blevi_sobolev.pdf