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Theorem 9.1. Let \( {X}_{2} \) be the cyclic double cover of \( {S}^{3} \) branched over a link \( L \) and suppose that \( A \) is a Seifert matrix for \( L \) with respect to some orientation and some Seifert surface. Then \( {H}_{1}\left( {X}_{2}\right) \) is presented, as an abelian group, by the matrix \( \left( {... | Proof. In the above notation, \( {\widehat{X}}_{2} = {Y}_{0} \cup {Y}_{1} \), where \( {Y}_{0} \cap {Y}_{1} \) is two disjoint copies of \( F \) . A presentation of \( {H}_{1}\left( {\widehat{X}}_{2}\right) \) can be obtained from the following exact Mayer-Vietoris sequence:\n\n\[ \rightarrow {H}_{1}\left( {{Y}_{0} \ca... | Yes |
Let \( {X}_{2} \) be the double cover of \( {S}^{3} \) branched over a link \( L \) . The order of the group \( {H}_{1}\left( {X}_{2}\right) \) is the modulus of the determinant of \( \left( {A + {A}^{\tau }}\right) \), that is\n\n\[ \left| {{H}_{1}\left( {X}_{2}\right) }\right| = \left| {\det \left( {A + {A}^{\tau }}\... | Proof. Any finitely generated abelian group can be expressed as a direct sum of cyclic groups. Thus it has as a presentation matrix a diagonal matrix, the entries on the diagonal being the orders of the summands, with the convention that an infinite group has order zero. By Theorem 6.1, the determinant of a square pres... | Yes |
Any Goeritz matrix for a link \( L \), associated with the white regions of a diagram of \( L \), represents, with respect to some base, the Gordon-Litherland form \[ {\mathcal{G}}_{F} : {H}_{1}\left( F\right) \times {H}_{1}\left( F\right) \rightarrow \mathbb{Z}, \] where \( F \) is the spanning surface for \( L \) giv... | Let the white regions, \( {R}_{0},{R}_{1},\ldots ,{R}_{n} \), of the diagram inherit an orientation from the sphere \( {S}^{2} \) in which they are assumed to lie; thus each \( \partial {R}_{i} \) has an orientation. Let \( {f}_{i} \) be the oriented simple closed curve in \( F \) that consists of \( \partial {R}_{i} \... | Yes |
Corollary 9.5. The determinant of \( L,\left| {{\Delta }_{L}\left( {-1}\right) }\right| \), is equal to \( \left| {\det G}\right| \), where \( G \) is any Goertiz matrix for \( L \) . | The proof of this is immediate from the last three theorems. It follows that \( \left| {\det G}\right| \) is an invariant of \( L \), and, as a Goeritz matrix is often easy to write down, it can be a useful invariant. | Yes |
Theorem 9.6. Suppose that \( {L}_{ + },{L}_{ - },{L}_{0} \) and \( {L}_{\infty } \) are four links that have identical diagrams except near a point where they are as shown in Figure 9.5. Then\n\n\[ \n{\left( \det {L}_{ + }\right) }^{2} + {\left( \det {L}_{ - }\right) }^{2} = 2\left( {{\left( \det {L}_{0}\right) }^{2} +... | Proof. The diagram shows the four links together with connected shaded spanning surfaces \( {F}_{i} \) for \( i = + , - ,0,\infty \) . These can always be constructed by using Seifert’s method (see Chapter 2) for \( {F}_{0} \) and adding bands to get the other three surfaces. The four surfaces are taken to be identical... | Yes |
Theorem 9.7. Let \( {X}_{r} \) be the cyclic \( r \) -fold cover of \( {S}^{3} \) branched over an \( n \) - component oriented link \( L \), and suppose that \( A \) is a Seifert matrix for \( L \) coming from a genus \( g \) Seifert surface. Then \( {H}_{1}\left( {X}_{r}\right) \) is presented, as an abelian group, b... | Assuming that a \ | No |
The order of the first homology group of \( {X}_{r} \), the cyclic \( r \) -fold cover of \( {S}^{3} \) branched over \( L \), is given by\n\n\[ \left| {{H}_{1}\left( {X}_{r}\right) }\right| = \left| {\mathop{\prod }\limits_{{v = 1}}^{{r - 1}}{\Delta }_{L}\left( {e}^{{2\pi }\imath \frac{v}{r}}\right) }\right| . \] | Assuming that a \ | No |
Lemma 10.5. Suppose that \( L \) and \( {L}^{\prime } \) are oriented links having property \( \left( \star \right) \) which are the same except near one point, where they are as shown in Figure 10.1; then \( \mathcal{A}\left( L\right) = \mathcal{A}\left( {L}^{\prime }\right) \) | Proof. The two segments shown on one of the two sides of Figure 10.1 must belong to the same component of the link. Suppose, without loss of generality, it is the two segments on the left side. Then using the Seifert circuit method of Theorem 2.2, a Seifert surface can be constructed for the left link that meets the ne... | Yes |
Theorem 10.7. Let \( K \) be a knot. Then \( \mathcal{A}\left( K\right) \equiv {a}_{2}\left( K\right) \) modulo 2, where \( {a}_{2}\left( K\right) \) is the coefficient of \( {z}^{2} \) in the Conway polynomial \( {\nabla }_{K}\left( z\right) \) . The Arfinvariant of \( K \) is related to the Alexander polynomial by\n\... | Proof. The formula \( \mathcal{A}\left( {L}_{ + }\right) - \mathcal{A}\left( {L}_{ - }\right) \equiv \operatorname{lk}\left( {L}_{0}\right) \) modulo 2, valid when \( {L}_{ + } \) has one component, allows calculation of \( \mathcal{A}\left( K\right) \) from \( \mathcal{A} \) (unknot) \( = 0 \) . However, this gives th... | Yes |
Theorem 11.1 (The Loop Theorem). Let \( M \) be a (possibly non-compact) 3- manifold with boundary \( \partial M \) such that the inclusion-induced homomorphism \( {\Pi }_{1}\left( {\partial M}\right) \rightarrow {\Pi }_{1}\left( M\right) \) is not injective. Then there exists a (piecewise linear) embedding of the disc... | Null | No |
Theorem 11.2. Let \( X \) be the exterior of a knot \( K \) in \( {S}^{3} \) . If \( K \) is not the unknot, then the inclusion map induces an injection \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow {\Pi }_{1}\left( X\right) \) . | Proof. Suppose \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow {\Pi }_{1}\left( X\right) \) is not injective. Then, by the loop theorem, there is an embedding \( e : {D}^{2} \rightarrow X \) sending \( \partial {D}^{2} \) into the torus \( \partial X \), to a simple closed curve not homotopically trivial in the tor... | Yes |
Corollary 11.3. A knot \( K \) is the unknot if and only if \( {\Pi }_{1}\left( {{S}^{3} - K}\right) \) is infinite cyclic. | Proof. If \( {\Pi }_{1}\left( {{S}^{3} - K}\right) \) is isomorphic to \( \mathbb{Z} \), there can be no injection \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow \) \( {\Pi }_{1}\left( X\right) \) (as \( {\Pi }_{1}\left( {\partial X}\right) \) is isomorphic to \( \mathbb{Z} \oplus \mathbb{Z} \) ). | No |
Corollary 11.4. Let \( {X}_{1} \) and \( {X}_{2} \) be the exteriors of two non-trivial knots and let M be a 3-manifold formed by identifying their boundaries together using any homeomorphism. Then the inclusion into \( M \) of the torus \( T \) that comes from the identified boundaries induces an injection \( {\Pi }_{... | Proof. This follows at once from the above theorem and from the Van Kam-pen theorem, which describes how fundamental groups behave when a space is described as a union of subspaces. | Yes |
Theorem 11.5 (The Sphere Theorem). Suppose that \( M \) is an orientable 3-manifold and that there exists a map \( {S}^{2} \rightarrow M \) that is not homotopic to a constant map (that is, \( {\Pi }_{2}\left( M\right) \neq 0 \) ). Then there exists a (piecewise linear) embedding \( {S}^{2} \rightarrow M \) that is not... | Null | No |
Theorem 11.6. If \( K \) is a knot in \( {S}^{3} \) any map \( {S}^{2} \rightarrow {S}^{3} - K \) is homotopic to a constant map (that is, \( {\Pi }_{2}\left( {{S}^{3} - K}\right) = 0 \) ). | Proof. If the statement is false then, by the sphere theorem, there exists a piecewise linear embedding \( e : {S}^{2} \rightarrow {S}^{3} - K \) that is not homotopic to a constant in \( \left( {{S}^{3} - K}\right) \) . Then, by the Schönflies theorem, \( e\left( {S}^{2}\right) \) separates \( {S}^{3} \) into two comp... | Yes |
Theorem 11.7. If \( K \) is a knot in \( {S}^{3} \), any map \( {S}^{r} \rightarrow {S}^{3} - K \) is homotopic to a constant map (that is, \( {\Pi }_{r}\left( {{S}^{3} - K}\right) = 0 \) ) for all \( r \geq 2 \) . | Proof. Let \( X \) be the exterior of \( K \) and let \( \widetilde{X} \) be the universal cover of \( X \) . Thus \( \widetilde{X} \) is the simply connected cover of \( X \), it is acted upon by \( {\Pi }_{1}\left( X\right) \), and the quotient of \( \widetilde{X} \) by this action is \( X \) . The operation of lifti... | Yes |
Theorem 11.8. If there exists an isomorphism from \( {\Pi }_{1}\left( {{S}^{3} - {K}_{1}}\right) \) to \( {\Pi }_{1}\left( {{S}^{3} - {K}_{2}}\right) \) which sends \( \left\lbrack {\lambda }_{1}\right\rbrack \) to \( \left\lbrack {\lambda }_{2}\right\rbrack \) and \( \left\lbrack {\mu }_{1}\right\rbrack \) and \( \lef... | Null | No |
Theorem 11.9. If \( {K}_{1} \) and \( {K}_{2} \) are prime knots in \( {S}^{3} \) and \( {\Pi }_{1}\left( {{S}^{3} - {K}_{1}}\right) \) and \( {\Pi }_{1}\left( {{S}^{3} - }\right. \) \( \left. {K}_{2}\right) \) are isomorphic groups, then \( \left( {{S}^{3} - {K}_{1}}\right) \) and \( \left( {{S}^{3} - {K}_{2}}\right) ... | Thus, for prime knots, the knot group determines the complement of the knot. It is by no means obvious that this means that the knots are the same. Perhaps the homeomorphism might send a meridian to a non-meridian. That this is not so is the substance of one of the most impressive results in knot theory of the 1980's. ... | No |
Theorem 11.10. If \( {K}_{1} \) and \( {K}_{2} \) are unoriented knots in \( {S}^{3} \) and there is an orientation preserving homeomorphism between their complements, then \( {K}_{1} \) and \( {K}_{2} \) are equivalent (as unoriented knots). | Null | No |
Lemma 12.2. Suppose that \( U \) and \( V \) are 3-manifolds with homeomorphic boundaries, and that \( {h}_{0} : \partial U \rightarrow \partial V \) and \( {h}_{1} : \partial U \rightarrow \partial V \) are isotopic homeomorphisms. Then \( U{ \cup }_{{h}_{0}}V \) and \( U{ \cup }_{{h}_{1}}V \) are homeomorphic. | Proof. Choose ([113],[47]) a collar neighbourhood \( C \) of \( \partial U \) in \( U \) ; \( C \) is a neighbourhood of \( \partial U \) homeomorphic to \( \partial U \times \left\lbrack {0,1}\right\rbrack \), with \( \partial U \) identified with \( \partial U \times 0 \) . A homeomorphism \( f : U{ \cup }_{{h}_{0}}V... | Yes |
Lemma 12.5. Suppose oriented simple closed curves \( p \) and \( q \), contained in the interior of the surface \( F \), intersect transversely at precisely one point. Then \( p{ \sim }_{\tau }q \) . | Proof. The first diagram of Figures 12.3 shows the intersection point of \( p \) and \( q \) and also a simple closed curve \( {C}_{1} \) that runs parallel to, and is slightly displaced from, \( q \) . Similarly, \( {C}_{2} \) is a slightly displaced copy of \( p \) . The second diagram shows \( {\tau }_{1}p \), where... | Yes |
Lemma 12.6. Suppose that oriented simple closed curves \( p \) and \( q \) contained in the interior of the surface \( F \) are disjoint and that neither separates \( F \) (that is, \( \left\lbrack p\right\rbrack \neq 0 \neq \left\lbrack q\right\rbrack \) in \( \left. {{H}_{1}\left( {F,\partial F}\right) }\right) \) . ... | Proof. Consideration of the surface obtained by cutting \( F \) along \( p \cup q \) shows at once that there is a simple closed curve \( r \) in \( F \) that intersects each of \( p \) and \( q \) transversely at one point. Then, by Lemma 12.5, \( p{ \sim }_{\tau }r{ \sim }_{\tau }q \) . | Yes |
Proposition 12.7. Suppose that oriented simple closed curves \( p \) and \( q \) are contained in the interior of the surface \( F \) and that neither separates \( F \) . Then \( p{ \sim }_{\tau }q \) . | Proof. Changing \( q \) by means of a homeomorphism of \( F \) that is (close to and) isotopic to the identity, it can be assumed that \( p \) and \( q \) intersect transversely at \( n \) points. The proof is by induction on \( n \) ; Lemmas 12.5 and 12.6 start the induction, so assume that \( n \geq 2 \) and that the... | Yes |
Corollary 12.8. Let \( {p}_{1},{p}_{2},\ldots ,{p}_{n} \) be disjoint simple closed curves in the interior of \( F \) the union of which does not separate \( F \) . Let \( {q}_{1},{q}_{2},\ldots ,{q}_{n} \) be another set of curves with the same properties. Then there is a homeomorphism \( h \) of \( F \) that is in th... | Proof. Suppose inductively that such an \( h \) can be found so that \( h{p}_{i} = {q}_{i} \) for each \( i = 1,2,\ldots, n - 1 \) . Apply Proposition 12.7 to \( h{p}_{n} \) and \( {q}_{n} \) in \( F \) cut along \( {q}_{1} \cup {q}_{2} \cup \ldots \cup {q}_{n - 1} \) | Yes |
Lemma 12.12. Any closed connected orientable 3-manifold has a Heegaard splitting. | Proof. This is similar to the first part of the proof of Theorem 8.2. Take a triangulation of \( M \) as a simplicial complex \( K \) . The vertices of the first derived subdivision \( {K}^{\left( 1\right) } \) of \( K \) are the barycentres \( \widehat{A} \) of the simplexes \( A \) of \( K \) . The second derived sub... | Yes |
Theorem 12.13. Let \( M \) be a closed connected orientable 3-manifold. There exists finite sets of disjoint solid tori \( {T}_{1}^{\prime },{T}_{2}^{\prime },\ldots ,{T}_{N}^{\prime } \) in \( M \) and \( {T}_{1},{T}_{2},\ldots ,{T}_{N} \) in \( {S}^{3} \) such that \( M - { \cup }_{1}^{N}\operatorname{Int}\left( {T}_... | Proof. By Lemma 12.12, \( M \) has a Heegaard splitting, so for handlebodies \( U \) and \( V \) of some genus \( g \), and some homeomorphism \( h : \partial U \rightarrow \partial V, M = \) \( U{ \cup }_{h}V \) . Let \( {p}_{1}^{\prime },{p}_{2}^{\prime },\ldots ,{p}_{g}^{\prime } \) be disjoint simple closed curves ... | No |
Theorem 12.14. Any closed connected orientable 3-manifold \( M \) can be obtained from \( {S}^{3} \) by a collection of 1 -surgeries, that is, by removing disjoint copies of \( {S}^{1} \times {D}^{2} \) and replacing them with copies of \( {D}^{2} \times {S}^{1} \) in the canonical way. Thus \( M \) bounds a 4-manifold... | Null | No |
Theorem 12.15. Two framed links in \( {S}^{3} \) give, by surgery, the same oriented 3- manifold if and only if they are related by a sequence of moves of two types. In a move of type 1, an extra unknotted component, unlinked from all other components, with framing 1 or -1 is added to or removed from the link. In a mov... | For the proof, which uses 4-dimensional Cerf theory, refer to [65]. If one considers the surgery information as a recipe for adding 2-handles on to a 4-ball to create a 4-manifold with the 3-manifold as its boundary, a move of type 2 corresponds to sliding one 2-handle over another. A type 1 move changes the 4-manifold... | No |
Lemma 13.2. Suppose that \( {A}^{4} \) is not a \( {k}^{\text{th }} \) root of unity for \( k \leq n \) . Then there is a unique element \( {f}^{\left( n\right) } \in T{L}_{n} \) such that\n\n(i) \( {f}^{\left( n\right) }{e}_{i} = 0 = {e}_{i}{f}^{\left( n\right) } \) for \( 1 \leq i \leq n - 1 \) ,\n\n(ii) \( \left( {{... | Proof. Note that if \( {f}^{\left( n\right) } \) exists, \( \mathbf{1} - {f}^{\left( n\right) } \) is the identity of the algebra generated by \( \left\{ {{e}_{1},{e}_{2},\ldots ,{e}_{n - 1}}\right\} \), and so \( {f}^{\left( n\right) } \) is then certainly unique. Let \( {f}^{\left( 0\right) } \) be the empty diagram ... | Yes |
Lemma 13.4. In \( \mathcal{S}\left( {{S}^{1} \times I,2\text{points}}\right) ,{a\omega } - {b\omega } \) is a linear sum of two elements, each of which contains a copy of \( {f}^{\left( r - 1\right) } \) . (That is, each of the two elements is the image of \( {f}^{\left( r - 1\right) } \) under some map \( T{L}_{r - 1}... | Proof. Consider the inclusion, shown in Figure 13.10, of the \( T{L}_{n + 1} \) recurrence relation of Figure 13.6 into the annulus.\n\n\n\nFigure 13.10\n\nThe top boundary points on either side of the square are joi... | Yes |
Lemma 13.5. Suppose that \( A \) is chosen so that \( {A}^{4} \) is a primitive \( {r}^{\text{th }} \) root of unity, \( r \geq 3 \) . Suppose that \( D \) is a planar diagram of a link of \( n \) (ordered) components. Suppose that \( {D}^{\prime } \) is another such diagram, obtained from \( D \) by a Kirby type 2 mov... | Proof. It must be checked that the elements of \( \mathcal{S}\left( {\mathbb{R}}^{2}\right) \), produced as described above from \( D \) and from \( {D}^{\prime } \), with \( \omega \) as the \ | No |
Corollary 13.6. If \( {A}^{4} \) is a primitive \( {r}^{\text{th }} \) root of unity, \( r \geq 3 \), and planar diagrams \( D \) and \( {D}^{\prime } \) are related by a sequence of Kirby moves of type 2, then\n\n\[< \omega ,\omega ,\ldots ,\omega { > }_{D} = < \omega ,\omega \ldots ,\omega { > }_{{D}^{\prime }}.\] | Null | No |
Theorem 13.8. Suppose that a closed oriented 3-manifold \( M \) is obtained by surgery on a framed link that is represented by a planar diagram D. Let \( {b}_{ + } \) be the number of positive eigenvalues and \( {b}_{ - } \) be the number negative eigenvalues of the linking matrix of this link. Suppose \( r \geq 3 \) a... | Proof. Note that \( A \) is a primitive \( 4{r}^{th} \) root of unity, and so, by Lemma 13.7, \( < \omega { > }_{{U}_{ + }} \) and \( < \omega { > }_{{U}_{ - }} \) are non-zero. It follows from the Corollary 13.6 and the preceding remarks about the linking matrix that the given expression is invariant under Kirby type ... | Yes |
Theorem 13.8. Suppose that a closed oriented 3-manifold \( M \) is obtained by surgery on a framed link that is represented by a planar diagram D. Let \( {b}_{ + } \) be the number of positive eigenvalues and \( {b}_{ - } \) be the number negative eigenvalues of the linking matrix of this link. Suppose \( r \geq 3 \) a... | Proof. Note that \( A \) is a primitive \( 4{r}^{th} \) root of unity, and so, by Lemma 13.7, \( < \omega { > }_{{U}_{ + }} \) and \( < \omega { > }_{{U}_{ - }} \) are non-zero. It follows from the Corollary 13.6 and the preceding remarks about the linking matrix that the given expression is invariant under Kirby type ... | Yes |
Lemma 13.9. Suppose \( r \geq 3 \) and \( A \) is a primitive \( 4{r}^{\text{th }} \) root of unity. The element of \( T{L}_{n} \) shown in Figure 13.13 is the zero map of outsides if \( 1 \leq n \leq r - 2 \) . When \( n = 0 \), the element acts as multiplication by \( \langle \omega {\rangle }_{U} \) . | Proof. Consider first the element of \( T{L}_{n} \) that consists of \( {f}^{\left( n\right) } \) encircled by one simple closed curve. This is shown in Figure 13.14 for \( n = 4 \) . Figure 13.14 shows a calculation for that element. Firstly one crossing is removed in the two standard ways, the results being multiplie... | Yes |
Lemma 14.2. The element of \( T{L}_{n} \) shown in Figure 14.2, which consists of the idempotent \( {f}^{\left( n\right) } \) with all its strands encircled by a parallel strands that join up the ends of an idempotent \( {f}^{\left( a\right) } \), is\n\n\[ \n{\left( -1\right) }^{a}\frac{{A}^{2\left( {n + 1}\right) \lef... | Proof. The \( a \) parallel strands and the idempotent \( {f}^{\left( a\right) } \) can, as explained in Chapter 13, be thought of as \( {S}_{a}\left( \alpha \right) \) contained in an annulus encircling the strands of \( {f}^{\left( n\right) } \), where \( {S}_{a} \) is the \( {a}^{\text{th }} \) Chebyshev polynomial.... | Yes |
Lemma 14.3. Suppose \( A \) is a primitive \( 4{r}^{\text{th }} \) root of unity. Then\n\n\[< \omega { > }_{{U}_{ + }} = \frac{G}{2{A}^{\left( 3 + {r}^{2}\right) }\left( {{A}^{2} - {A}^{-2}}\right) },\] | Proof. Recall that \( {U}_{ + } \) is the diagram of the unknot with one positive crossing,\n\n\[ \omega = \mathop{\sum }\limits_{{n = 0}}^{{r - 2}}{\Delta }_{n}{S}_{n}\left( \alpha \right) \text{ and }{\Delta }_{n} = \frac{{\left( -1\right) }^{n}\left( {{A}^{2\left( {n + 1}\right) } - {A}^{-2\left( {n + 1}\right) }}\r... | Yes |
Lemma 14.5.\n\n\\[ \n\\Gamma \\left( {x, y, z}\\right) = \\frac{{\\Delta }_{x + y + z}!{\\Delta }_{x - 1}!{\\Delta }_{y - 1}!{\\Delta }_{z - 1}!}{{\\Delta }_{y + z - 1}!{\\Delta }_{z + x - 1}!{\\Delta }_{x + y - 1}!}. \n\\] | Proof. Consider the equations depicted in Figure 14.4; as usual a symbol beside a line is a count of the number of parallel arcs that it represents. The first equality follows from the defining relation of Figure 13.6 for \\( {f}^{\\left( y + z - 1\\right) } \\) (together with \\( {f}^{\\left( z\\right) }{e}_{z - 1} = ... | Yes |
Lemma 14.7. Let \( \left( {a, b, c}\right) \) be admissible and let \( A \) be a primitive \( 4{r}^{\text{th }} \) root of unity. Then \( {\tau }_{a, b, c}^{ * } \) is non-zero if and only if \( a + b + c \leq 2\left( {r - 2}\right) \) . | Proof. \( \mathcal{S}{D}^{\prime } \) has a base consisting of all diagrams in \( {D}^{\prime } \) with no crossing. For all but one of these diagrams there is an arc from a point of one of the three specified subsets (for example, that with \( a \) points) to another point of the same subset. As usual (using \( {f}^{\... | Yes |
Lemma 14.9. Suppose \( A \) is not a root of unity. \( A \) base for \( {Q}_{a, b, c, d} \) is the set of elements as in Figure 14.10 (the boundary of the disc is not shown), where \( j \) takes all values such that \( \left( {a, b, j}\right) \) and \( \left( {c, d, j}\right) \) are both admissible. | Proof. Note that the proposed base elements each consist of two triads glued together; there is an \( {f}^{\left( j\right) } \) on the central line. Certainly \( {Q}_{a, b, c, d} \) is spanned by all elements of the form shown in Figure 14.11, where the lines all represent multiple parallel arcs, for, as usual, any oth... | Yes |
Lemma 14.10. Suppose \( A \) is a primitive \( 4{r}^{\text{th }} \) root of unity. A base for \( {Q}_{a, b, c, d}^{ * } \) (this being \( {Q}_{a, b, c, d} \) regarded as maps of diagrams outside the disc) is the set of elements as in Figure 14.10 where \( j \) takes all values such that \( \left( {a, b, j}\right) \) an... | Proof. The proof that the given elements span is the same as in Lemma 14.9 with a small modification. Now, \( {f}^{\left( n\right) } \) does not exist for \( n \geq r \) . However, \( {f}^{\left( r - 1\right) } \) is the zero map of outsides. Thus working in this dual context, any diagram as in the above proof, with at... | Yes |
Lemma 14.11. In \( \mathcal{S}\left( {{S}^{1} \times I}\right) ,{S}_{a}\left( \alpha \right) {S}_{b}\left( \alpha \right) = \mathop{\sum }\limits_{c}{S}_{c}\left( \alpha \right) \) where the summation is over all \( c \) such that \( \left( {a, b, c}\right) \) is admissible. If \( \bar{A} \) is a primitive \( 4{r}^{\te... | Proof. In fact the first part of this lemma is almost immediate. This is because it is a result on Chebyshev polynomials that \( {S}_{a}\left( x\right) {S}_{b}\left( x\right) = \mathop{\sum }\limits_{c}{S}_{c}\left( x\right) \), the sum being over all \( c \) such that \( \left( {a, b, c}\right) \) is admissible. This ... | Yes |
Theorem 14.12. Let \( {F}_{g} \) be the closed orientable surface of genus \( g \) and let \( A \) be a primitive \( 4{r}^{\text{th }} \) root of unity, \( r \geq 3 \) . Then \( {\mathcal{I}}_{A}\left( {{S}^{1} \times {F}_{g}}\right) \) is an integer. It is \( r - 1 \) when \( g = 1 \) . Otherwise it is the number of w... | Proof. The 3-manifold \( {S}^{1} \times {F}_{g} \) is obtained by surgery on a link that consists of \( g \) copies of the Borromean rings summed together on one component, each component having the zero framing. (Proving this is an interesting exercise.) A diagram \( D \) for such a link is obtained by taking \( g \) ... | Yes |
Theorem 14.13. If \( p \) and \( q \) are coprime positive integers, then the Jones polynomial of the \( \left( {p, q}\right) \) -torus knot is\n\n\[ \n{t}^{\left( {p - 1}\right) \left( {q - 1}\right) /2}{\left( 1 - {t}^{2}\right) }^{-1}\left( {1 - {t}^{p + 1} - {t}^{q + 1} + {t}^{p + q}}\right) .\n\] | Proof. Consider the diagram of Figure 14.23, which shows \( p \) arcs traversing a rectangle. Suppose \( q \) copies of this are placed side by side and the result is closed up by joining the \( p \) points on the left to those on the right, using \( p \) crossing-free arcs encircling an annulus \( {S}^{1} \times I \) ... | Yes |
Lemma 15.1. Suppose that \( p \) and \( q \) are two arcs in \( {\mathbb{R}}^{2} \) meeting only at their end points \( A \) and \( B \), and let \( R \) be the compact region bounded by \( p \cup q \) . Suppose that \( {t}_{1},{t}_{2},\ldots {t}_{n} \) are arcs in \( R \), each meeting \( p \cup q \) at just its end p... | Proof. Proceed by induction on the number \( n \) of arcs. The result is trivial if \( n = 1 \), so assume \( n > 1 \) . Amongst the end points of the \( {t}_{i} \) that lie on \( p \), let \( X \) be the nearest to \( A \) . If then \( X \) is an end of \( {t}_{j} \), let \( {B}^{\prime } \) be the other end of \( {t}... | Yes |
Theorem 15.5. There exists a function \[ \Lambda : \left\{ {\text{ Unoriented links diagrams in }{S}^{2}}\right\} \rightarrow \mathbb{Z}\left\lbrack {{a}^{\pm 1},{z}^{\pm 1}}\right\rbrack \] that is defined uniquely by the following: (i) \( \Lambda \left( U\right) = 1 \), where \( U \) is the zero-crossing diagram of t... | Proof. Note that, when considering a crossing in an unoriented diagram, it has no claim to be termed \( {D}_{ + } \) rather than \( {D}_{ - } \) in the above notation. However, this never matters, since \( {D}_{ + } \) and \( {D}_{ - } \) feature symmetrically in the formula \( \left( {\star \star }\right) \) ; the tre... | Yes |
Proposition 16.2. If \( {L}_{1} \) and \( {L}_{2} \) are oriented links, then\n\n(i) \( P\left( {{L}_{1} + {L}_{2}}\right) = P\left( {L}_{1}\right) P\left( {L}_{2}\right) \) ;\n\n(ii) \( F\left( {{L}_{1} + {L}_{2}}\right) = F\left( {L}_{1}\right) F\left( {L}_{2}\right) \) ;\n\n(iii) \( P\left( {{L}_{1} \sqcup {L}_{2}}\... | Null | No |
Proposition 16.3. Both \( P\left( L\right) \) and \( F\left( L\right) \) are unchanged by mutation of \( L \) . | Null | No |
Proposition 16.4. If the oriented link \( {L}^{ * } \) is obtained from the oriented link \( L \) by reversing the orientation of one component \( K \), then\n\n\[ F\left( {L}^{ * }\right) = {a}^{4\operatorname{lk}\left( {K, L - K}\right) }F\left( L\right) . \]\n\nChanging the orientation of all components of \( L \) l... | Null | No |
Proposition 16.5. For an oriented link \( L \), the Conway-normalised Alexander polynomial \( {\Delta }_{L}\left( t\right) \) and the Jones polynomial \( V\left( L\right) \) are related to the HOMFLY polynomial \( P\left( L\right) \) by \[ {\Delta }_{L}\left( t\right) = P{\left( L\right) }_{\left( i, i\left( {t}^{1/2} ... | Null | No |
Proposition 16.6. For an oriented link \( L \) , \[ V\left( L\right) = F\left( L\right) \text{ when }\left( {a, z}\right) = \left( {-{t}^{-3/4},\;\left( {{t}^{-1/4} + {t}^{1/4}}\right) }\right) , \] \[ {\left( V\left( L\right) \right) }^{2} = {\left( -1\right) }^{\# L - 1}F\left( L\right) \text{ when }t = - {q}^{-2},\;... | Proof. Underlying the Jones polynomial is the Kauffman bracket. Reverting to the notation for that, given in Definition 3.1, \[ \langle > < \rangle = A\langle > < \rangle + {A}^{-1}\langle < \rangle \] \[ \langle > < \rangle = {A}^{-1}\langle > < \rangle + A\langle > < \rangle . \] Adding these equations gives \[ \lang... | Yes |
The \( l \) -breadth of \( P\left( L\right) \) satisfies \( {E}_{l}\left( {P\left( L\right) }\right) - {e}_{l}\left( {P\left( L\right) }\right) \leq \) \( 2\left( {s\left( D\right) - 1}\right) \) . | Null | No |
Proposition 16.9. Suppose \( L \) is an oriented link with \( \# L \) components. Then \( (1 - \) \( \# L) \) is the lowest power both of \( m \) in \( P\left( L\right) \) and of \( z \) in \( F\left( L\right) \), and | \[ {\left\lbrack {z}^{\# L - 1}F\left( L\right) \right\rbrack }_{\left( {a, z}\right) = \left( {l,0}\right) } = {\left\lbrack {\left( -m\right) }^{\# L - 1}P\left( L\right) \right\rbrack }_{m = 0}. \] | Yes |
Proposition 16.10. Any oriented link in \( {S}^{3} \) is the closure \( \widehat{\xi } \) of some \( \xi \) belonging to the braid group \( {B}_{n} \), for some \( n \) . Oriented links \( \widehat{\xi } \) and \( \widehat{\eta } \) are equivalent if \( \xi \) and \( \eta \) differ by a sequence of (Markov) moves of th... | Null | No |
Theorem 16.11. If an oriented link \( L \) is the closure of \( \xi \in {B}_{n} \), let \( T\left( L\right) \) be defined to be \( T\left( \xi \right) \) . This is a well-defined link invariant. | Proof. Because \( T \) is essentially a trace function, if \( \xi ,\eta \in {B}_{n} \) then \( T\left( {{\eta }^{-1}{\xi \eta }}\right) = \n\n\( T\left( \xi \right) \) . Using the properties of \( \mu \), it is easy to show that \( T\left( {\xi {\sigma }_{n}}\right) = T\left( {\xi {\sigma }_{n}^{-1}}\right) = \n\n\( T\... | No |
Theorem 2.1 Suppose that \( f \) is an integrable function on the circle with \( \widehat{f}\left( n\right) = 0 \) for all \( n \in \mathbb{Z} \) . Then \( f\left( {\theta }_{0}\right) = 0 \) whenever \( f \) is continuous at the point \( {\theta }_{0} \) . | Proof. We suppose first that \( f \) is real-valued, and argue by contradiction. Assume, without loss of generality, that \( f \) is defined on \( \left\lbrack {-\pi ,\pi }\right\rbrack \), that \( {\theta }_{0} = 0 \), and \( f\left( 0\right) > 0 \) . The idea now is to construct a family of trigonometric polynomials ... | No |
Corollary 2.2 If \( f \) is continuous on the circle and \( \widehat{f}\left( n\right) = 0 \) for all \( n \in \mathbb{Z} \), then \( f = 0 \) . | Null | No |
Corollary 2.3 Suppose that \( f \) is a continuous function on the circle and that the Fourier series of \( f \) is absolutely convergent, \( \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\left| {\widehat{f}\left( n\right) }\right| < \infty \) . Then, the Fourier series converges uniformly to \( f \), that is,\n\n\[... | Proof. Recall that if a sequence of continuous functions converges uniformly, then the limit is also continuous. Now observe that the assumption \( \sum \left| {\widehat{f}\left( n\right) }\right| < \infty \) implies that the partial sums of the Fourier\n\nseries of \( f \) converge absolutely and uniformly, and theref... | Yes |
Corollary 2.4 Suppose that \( f \) is a twice continuously differentiable function on the circle. Then\n\n\[ \widehat{f}\left( n\right) = O\left( {1/{\left| n\right| }^{2}}\right) \;\text{ as }\left| n\right| \rightarrow \infty ,\] \n\nso that the Fourier series of \( f \) converges absolutely and uniformly to \( f \) ... | Proof. The estimate on the Fourier coefficients is proved by integrating by parts twice for \( n \neq 0 \) . We obtain\n\n\[ {2\pi }\widehat{f}\left( n\right) = {\int }_{0}^{2\pi }f\left( \theta \right) {e}^{-{in\theta }}{d\theta } \]\n\n\[ = {\left\lbrack f\left( \theta \right) \cdot \frac{-{e}^{-{in\theta }}}{in}\rig... | Yes |
Proposition 3.1 Suppose that \( f, g \), and \( h \) are \( {2\pi } \) -periodic integrable functions. Then:\n\n(i) \( f * \left( {g + h}\right) = \left( {f * g}\right) + \left( {f * h}\right) \) . | Proof. Properties (i) and (ii) follow at once from the linearity of the integral. | Yes |
Lemma 3.2 Suppose \( f \) is integrable on the circle and bounded by \( B \) . Then there exists a sequence \( {\left\{ {f}_{k}\right\} }_{k = 1}^{\infty } \) of continuous functions on the circle so that\n\n\[ \mathop{\sup }\limits_{{x \in \left\lbrack {-\pi ,\pi }\right\rbrack }}\left| {{f}_{k}\left( x\right) }\right... | Using this result, we may complete the proof of the proposition as follows. Apply Lemma 3.2 to \( f \) and \( g \) to obtain sequences \( \left\{ {f}_{k}\right\} \) and \( \left\{ {g}_{k}\right\} \) of approximating continuous functions. Then\n\n\[ f * g - {f}_{k} * {g}_{k} = \left( {f - {f}_{k}}\right) * g + {f}_{k} *... | Yes |
Theorem 4.1 Let \( {\left\{ {K}_{n}\right\} }_{n = 1}^{\infty } \) be a family of good kernels, and \( f \) an integrable function on the circle. Then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {f * {K}_{n}}\right) \left( x\right) = f\left( x\right) \]\n\nwhenever \( f \) is continuous at \( x \) . If ... | Proof of Theorem 4.1. If \( \epsilon > 0 \) and \( f \) is continuous at \( x \), choose \( \delta \) so that \( \left| y\right| < \delta \) implies \( \left| {f\left( {x - y}\right) - f\left( x\right) }\right| < \epsilon \) . Then, by the first property of good kernels, we can write\n\n\[ \left( {f * {K}_{n}}\right) \... | Yes |
Lemma 5.1 We have\n\n\[ \n{F}_{N}\left( x\right) = \frac{1}{N}\frac{{\sin }^{2}\left( {{Nx}/2}\right) }{{\sin }^{2}\left( {x/2}\right) }\n\]\n\nand the Fejér kernel is a good kernel. | The proof of the formula for \( {F}_{N} \) (a simple application of trigonometric identities) is outlined in Exercise 15. To prove the rest of the lemma, note that \( {F}_{N} \) is positive and \( \frac{1}{2\pi }{\int }_{-\pi }^{\pi }{F}_{N}\left( x\right) {dx} = 1 \), in view of the fact that a similar identity holds ... | No |
Theorem 5.2 If \( f \) is integrable on the circle, then the Fourier series of \( f \) is Cesàro summable to \( f \) at every point of continuity of \( f \) . | Null | No |
Corollary 5.3 If \( f \) is integrable on the circle and \( \widehat{f}\left( n\right) = 0 \) for all \( n \) , then \( f = 0 \) at all points of continuity of \( f \) . | The proof is immediate since all the partial sums are 0 , hence all the Cesàro means are 0 . | Yes |
Corollary 5.4 Continuous functions on the circle can be uniformly approximated by trigonometric polynomials. | This means that if \( f \) is continuous on \( \left\lbrack {-\pi ,\pi }\right\rbrack \) with \( f\left( {-\pi }\right) = f\left( \pi \right) \) and \( \epsilon > 0 \), then there exists a trigonometric polynomial \( P \) such that\n\n\[ \left| {f\left( x\right) - P\left( x\right) }\right| < \epsilon \;\text{ for all }... | Yes |
Lemma 5.5 If \( 0 \leq r < 1 \), then\n\n\[ \n{P}_{r}\left( \theta \right) = \frac{1 - {r}^{2}}{1 - {2r}\cos \theta + {r}^{2}}.\n\] | Proof. The identity \( {P}_{r}\left( \theta \right) = \frac{1 - {r}^{2}}{1 - {2r}\cos \theta + {r}^{2}} \) has already been derived in Section 1.1. Note that\n\n\[ \n1 - {2r}\cos \theta + {r}^{2} = {\left( 1 - r\right) }^{2} + {2r}\left( {1 - \cos \theta }\right) .\n\]\n\nHence if \( 1/2 \leq r \leq 1 \) and \( \delta ... | Yes |
Theorem 5.6 The Fourier series of an integrable function on the circle is Abel summable to \( f \) at every point of continuity. Moreover, if \( f \) is continuous on the circle, then the Fourier series of \( f \) is uniformly Abel summable to \( f \) . | Null | No |
Theorem 1.1 Suppose \( f \) is integrable on the circle. Then\n\n\[ \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( \theta \right) - {S}_{N}\left( f\right) \left( \theta \right) \right| }^{2}{d\theta } \rightarrow 0\;\text{ as }N \rightarrow \infty . \] | Null | No |
Lemma 1.2 (Best approximation) If \( f \) is integrable on the circle with Fourier coefficients \( {a}_{n} \), then\n\n\[ \begin{Vmatrix}{f - {S}_{N}\left( f\right) }\end{Vmatrix} \leq \begin{Vmatrix}{f - \mathop{\sum }\limits_{{\left| n\right| \leq N}}{c}_{n}{e}_{n}}\end{Vmatrix} \]\n\nfor any complex numbers \( {c}_{... | Proof. This follows immediately by applying the Pythagorean theorem to\n\n\[ f - \mathop{\sum }\limits_{{\left| n\right| \leq N}}{c}_{n}{e}_{n} = f - {S}_{N}\left( f\right) + \mathop{\sum }\limits_{{\left| n\right| \leq N}}{b}_{n}{e}_{n} \]\n\nwhere \( {b}_{n} = {a}_{n} - {c}_{n} \) . | Yes |
Theorem 1.3 Let \( f \) be an integrable function on the circle with \( f \sim \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n}{e}^{in\theta } \) . Then we have:\n\n(i) Mean-square convergence of the Fourier series\n\n\[ \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( \theta \right) - {S}_{N}\left( f\right) \... | Null | No |
Theorem 1.4 (Riemann-Lebesgue lemma) If \( f \) is integrable on the circle, then \( \widehat{f}\left( n\right) \rightarrow 0 \) as \( \left| n\right| \rightarrow \infty \) . | Null | No |
Lemma 1.5 Suppose \( F \) and \( G \) are integrable on the circle with\n\n\[ F \sim \sum {a}_{n}{e}^{in\theta }\;\text{ and }\;G \sim \sum {b}_{n}{e}^{in\theta }.\]\n\nThen\n\n\[ \frac{1}{2\pi }{\int }_{0}^{2\pi }F\left( \theta \right) \overline{G\left( \theta \right) }{d\theta } = \mathop{\sum }\limits_{{n = - \infty... | Proof. The proof follows from Parseval's identity and the fact that\n\n\[ \left( {F, G}\right) = \frac{1}{4}\left\lbrack {\parallel F + G{\parallel }^{2} - \parallel F - G{\parallel }^{2} + i\left( {\parallel F + {iG}{\parallel }^{2} - \parallel F - {iG}{\parallel }^{2}}\right) }\right\rbrack \]\n\nwhich holds in every... | No |
Theorem 2.1 Let \( f \) be an integrable function on the circle which is differentiable at a point \( {\theta }_{0} \) . Then \( {S}_{N}\left( f\right) \left( {\theta }_{0}\right) \rightarrow f\left( {\theta }_{0}\right) \) as \( N \) tends to infinity. | Proof. Define\n\n\[ F\left( t\right) = \left\{ \begin{array}{ll} \frac{f\left( {{\theta }_{0} - t}\right) - f\left( {\theta }_{0}\right) }{t} & \text{ if }t \neq 0\text{ and }\left| t\right| < \pi \\ - {f}^{\prime }\left( {\theta }_{0}\right) & \text{ if }t = 0. \end{array}\right. \]\n\nFirst, \( F \) is bounded near 0... | Yes |
Theorem 2.2 Suppose \( f \) and \( g \) are two integrable functions defined on the circle, and for some \( {\theta }_{0} \) there exists an open interval \( I \) containing \( {\theta }_{0} \) such that\n\n\[ f\left( \theta \right) = g\left( \theta \right) \;\text{ for all }\theta \in I. \]\n\nThen \( {S}_{N}\left( f\... | Proof. The function \( f - g \) is 0 in \( I \), so it is differentiable at \( {\theta }_{0} \), and we may apply the previous theorem to conclude the proof. | Yes |
Lemma 2.3 Suppose that the Abel means \( {A}_{r} = \mathop{\sum }\limits_{{n = 1}}^{\infty }{r}^{n}{c}_{n} \) of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{c}_{n} \) are bounded as \( r \) tends to 1 (with \( r < 1 \) ). If \( {c}_{n} = O\left( {1/n}\right) \), then the partial sums \( {S}_{N} = \mathop{\s... | Proof. Let \( r = 1 - 1/N \) and choose \( M \) so that \( n\left| {c}_{n}\right| \leq M \) . We estimate the difference\n\n\[ \n{S}_{N} - {A}_{r} = \mathop{\sum }\limits_{{n = 1}}^{N}\left( {{c}_{n} - {r}^{n}{c}_{n}}\right) - \mathop{\sum }\limits_{{n = N + 1}}^{\infty }{r}^{n}{c}_{n} \n\]\n\nas follows:\n\n\[ \n\left... | Yes |
Lemma 2.4\n\n\[ \n{S}_{M}\left( {P}_{N}\right) = \left\{ \begin{array}{ll} {P}_{N} & \text{ if }M \geq {3N} \\ {\widetilde{P}}_{N} & \text{ if }M = {2N} \\ 0 & \text{ if }M < N \end{array}\right. \n\] | This is clear from what has been said above and from Figure 3. | No |
Lemma 2.2 If \( f \) is continuous and periodic of period 1, and \( \gamma \) is irrational, then\n\n\[ \n\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}f\left( {n\gamma }\right) \rightarrow {\int }_{0}^{1}f\left( x\right) {dx}\;\text{ as }N \rightarrow \infty .\n\] | The proof of the lemma is divided into three steps.\n\nStep 1. We first check the validity of the limit in the case when \( f \) is one of the exponentials \( 1,{e}^{2\pi ix},\ldots ,{e}^{2\pi ikx},\ldots \) . If \( f = 1 \), the limit\nsurely holds. If \( f = {e}^{2\pi ikx} \) with \( k \neq 0 \), then the integral is... | Yes |
Corollary 2.3 The conclusion of Lemma 2.2 holds for every function \( f \) which is Riemann integrable in \( \left\lbrack {0,1}\right\rbrack \), and periodic of period 1 . | Proof. Assume \( f \) is real-valued, and consider a partition of the interval \( \left\lbrack {0,1}\right\rbrack \), say \( 0 = {x}_{0} < {x}_{1} < \cdots < {x}_{N} = 1 \) . Next, define \( {f}_{U}\left( x\right) = \) \( \mathop{\sup }\limits_{{{x}_{j - 1} \leq y \leq {x}_{j}}}f\left( y\right) \) if \( x \in \left\lbr... | Yes |
Lemma 3.3 If \( {2N} = {2}^{n} \), then\n\n\[{\bigtriangleup }_{2N}\left( f\right) - {\bigtriangleup }_{N}\left( f\right) = {2}^{-{n\alpha }}{e}^{i{2}^{n}x}.\] | This follows from our previous observation (6) because \( {\bigtriangleup }_{2N}\left( f\right) = \) \( {S}_{2N}\left( f\right) \) and \( {\bigtriangleup }_{N}\left( f\right) = {S}_{N}\left( f\right) \) . | No |
Proposition 1.1 The integral of a function of moderate decrease defined by (5) satisfies the following properties:\n\n(i) Linearity: if \( f, g \in \mathcal{M}\left( \mathbb{R}\right) \) and \( a, b \in \mathbb{C} \), then\n\n\[ \n{\int }_{-\infty }^{\infty }\left( {{af}\left( x\right) + {bg}\left( x\right) }\right) {d... | We say a few words about the proof. Property (i) is immediate. | No |
Proposition 1.2 If \( f \in \mathcal{S}\left( \mathbb{R}\right) \) then:\n\n(i) \( f\left( {x + h}\right) \rightarrow \widehat{f}\left( \xi \right) {e}^{2\pi ih\xi } \) whenever \( h \in \mathbb{R} \) .\n\n(ii) \( f\left( x\right) {e}^{-{2\pi ixh}} \rightarrow \widehat{f}\left( {\xi + h}\right) \) whenever \( h \in \ma... | Proof. Property (i) is an immediate consequence of the translation invariance of the integral, and property (ii) follows from the definition. Also, the third property of Proposition 1.1 establishes (iii).\n\nIntegrating by parts gives\n\n\[{\int }_{-N}^{N}{f}^{\prime }\left( x\right) {e}^{-{2\pi ix\xi }}{dx} = {\left\l... | Yes |
Theorem 1.3 If \( f \in \mathcal{S}\left( \mathbb{R}\right) \), then \( \widehat{f} \in \mathcal{S}\left( \mathbb{R}\right) \) . | The proof is an easy application of the fact that the Fourier transform interchanges differentiation and multiplication. In fact, note that if \( f \in \) \( \mathcal{S}\left( \mathbb{R}\right) \), its Fourier transform \( \widehat{f} \) is bounded; then also, for each pair of non-negative integers \( \ell \) and \( k ... | Yes |
Theorem 1.4 If \( f\left( x\right) = {e}^{-\pi {x}^{2}} \), then \( \widehat{f}\left( \xi \right) = f\left( \xi \right) \) . | Proof. Define\n\n\[ F\left( \xi \right) = \widehat{f}\left( \xi \right) = {\int }_{-\infty }^{\infty }{e}^{-\pi {x}^{2}}{e}^{-{2\pi ix\xi }}{dx} \]\n\nand observe that \( F\left( 0\right) = 1 \), by our previous calculation. By property (v) in Proposition 1.2, and the fact that \( {f}^{\prime }\left( x\right) = - {2\pi... | Yes |
Corollary 1.5 If \( \delta > 0 \) and \( {K}_{\delta }\left( x\right) = {\delta }^{-1/2}{e}^{-\pi {x}^{2}/\delta } \), then \( \widehat{{K}_{\delta }}\left( \xi \right) = {e}^{-{\pi \delta }{\xi }^{2}} \) . | Null | No |
Theorem 1.6 The collection \( {\left\{ {K}_{\delta }\right\} }_{\delta > 0} \) is a family of good kernels as \( \delta \rightarrow 0 \) . | Proof. First, we claim that \( f \) is uniformly continuous on \( \mathbb{R} \) . Indeed, given \( \epsilon > 0 \) there exists \( R > 0 \) so that \( \left| {f\left( x\right) }\right| < \epsilon /4 \) whenever \( \left| x\right| \geq R \) . Moreover, \( f \) is continuous, hence uniformly continuous on the compact int... | Yes |
Corollary 1.7 If \( f \in \mathcal{S}\left( \mathbb{R}\right) \), then\n\n\[ \left( {f * {K}_{\delta }}\right) \left( x\right) \rightarrow f\left( x\right) \;\text{ uniformly in }x\text{ as }\delta \rightarrow 0. \] | Proof. First, we claim that \( f \) is uniformly continuous on \( \mathbb{R} \) . Indeed, given \( \epsilon > 0 \) there exists \( R > 0 \) so that \( \left| {f\left( x\right) }\right| < \epsilon /4 \) whenever \( \left| x\right| \geq R \) . Moreover, \( f \) is continuous, hence uniformly continuous on the compact int... | Yes |
Proposition 1.8 If \( f, g \in \mathcal{S}\left( \mathbb{R}\right) \), then\n\n\[{\int }_{-\infty }^{\infty }f\left( x\right) \widehat{g}\left( x\right) {dx} = {\int }_{-\infty }^{\infty }\widehat{f}\left( y\right) g\left( y\right) {dy}.\] | To prove the proposition, we need to digress briefly to discuss the interchange of the order of integration for double integrals. Suppose \( F\left( {x, y}\right) \) is a continuous function in the plane \( \left( {x, y}\right) \in {\mathbb{R}}^{2} \) . We will assume the following decay condition on \( F \) :\n\n\[ \l... | Yes |
Theorem 1.9 (Fourier inversion) If \( f \in \mathcal{S}\left( \mathbb{R}\right) \), then\n\n\[ f\left( x\right) = {\int }_{-\infty }^{\infty }\widehat{f}\left( \xi \right) {e}^{2\pi ix\xi }{d\xi } \] | Proof. We first claim that\n\n\[ f\left( 0\right) = {\int }_{-\infty }^{\infty }\widehat{f}\left( \xi \right) {d\xi } \]\n\nLet \( {G}_{\delta }\left( x\right) = {e}^{-{\pi \delta }{x}^{2}} \) so that \( \widehat{{G}_{\delta }}\left( \xi \right) = {K}_{\delta }\left( \xi \right) \). By the multiplication formula we get... | Yes |
Corollary 1.10 The Fourier transform is a bijective mapping on the Schwartz space. | Null | No |
Proposition 1.11 If \( f, g \in \mathcal{S}\left( \mathbb{R}\right) \) then:\n\n(i) \( f * g \in \mathcal{S}\left( \mathbb{R}\right) \) .\n\n(ii) \( f * g = g * f \) .\n\n(iii) \( \widehat{\left( f * g\right) }\left( \xi \right) = \widehat{f}\left( \xi \right) \widehat{g}\left( \xi \right) \) . | Proof. To prove that \( f * g \) is rapidly decreasing, observe first that for any \( \ell \geq 0 \) we have \( \mathop{\sup }\limits_{x}{\left| x\right| }^{\ell }\left| {g\left( {x - y}\right) }\right| \leq {A}_{\ell }{\left( 1 + \left| y\right| \right) }^{\ell } \), because \( g \) is rapidly decreasing (to check thi... | Yes |
Theorem 1.12 (Plancherel) If \( f \in \mathcal{S}\left( \mathbb{R}\right) \) then \( \parallel \widehat{f}\parallel = \parallel f\parallel \) . | Proof. If \( f \in \mathcal{S}\left( \mathbb{R}\right) \) define \( {f}^{b}\left( x\right) = \overline{f\left( {-x}\right) } \) . Then \( \widehat{{f}^{b}}\left( \xi \right) = \overline{\widehat{f}\left( \xi \right) } \) . Now let \( h = f * {f}^{b} \) . Clearly, we have\n\n\[ \n\widehat{h}\left( \xi \right) = {\left| ... | Yes |
Corollary 2.2 \( u\left( {\cdot, t}\right) \) belongs to \( \mathcal{S}\left( \mathbb{R}\right) \) uniformly in \( t \), in the sense that for any \( T > 0 \)\n\n(9)\n\n\[ \mathop{\sup }\limits_{\substack{{x \in \mathbb{R}} \\ {0 < t < T} }}{\left| x\right| }^{k}\left| {\frac{{\partial }^{\ell }}{\partial {x}^{\ell }}u... | Proof. This result is a consequence of the following estimate:\n\n\[ \left| {u\left( {x, t}\right) }\right| \leq {\int }_{\left| y\right| \leq \left| x\right| /2}\left| {f\left( {x - y}\right) }\right| {\mathcal{H}}_{t}\left( y\right) {dy} + {\int }_{\left| y\right| \geq \left| x\right| /2}\left| {f\left( {x - y}\right... | Yes |
Theorem 2.3 Suppose \( u\left( {x, t}\right) \) satisfies the following conditions:\n\n(i) \( u \) is continuous on the closure of the upper half-plane.\n\n(ii) \( u \) satisfies the heat equation for \( t > 0 \) .\n\n(iii) \( u \) satisfies the boundary condition \( u\left( {x,0}\right) = 0 \) .\n\n(iv) \( u\left( {\c... | Proof. We define the energy at time \( t \) of the solution \( u\left( {x, t}\right) \) by\n\n\[ E\left( t\right) = {\int }_{\mathbb{R}}{\left| u\left( x, t\right) \right| }^{2}{dx} \]\n\nClearly \( E\left( t\right) \geq 0 \) . Since \( E\left( 0\right) = 0 \) it suffices to show that \( E \) is a decreasing function, ... | Yes |
Lemma 2.4 The following two identities hold:\n\n\[ \n{\int }_{-\infty }^{\infty }{e}^{-{2\pi }\left| \xi \right| y}{e}^{2\pi i\xi x}{d\xi } = {\mathcal{P}}_{y}\left( x\right) \n\]\n\n\[ \n{\int }_{-\infty }^{\infty }{\mathcal{P}}_{y}\left( x\right) {e}^{-{2\pi ix\xi }}{dx} = {e}^{-{2\pi }\left| \xi \right| y}. \n\] | Proof. The first formula is fairly straightforward since we can split the integral from \( - \infty \) to 0 and 0 to \( \infty \) . Then, since \( y > 0 \) we have\n\n\[ \n{\int }_{0}^{\infty }{e}^{-{2\pi \xi y}}{e}^{2\pi i\xi x}{d\xi } = {\int }_{0}^{\infty }{e}^{{2\pi i}\left( {x + {iy}}\right) \xi }{d\xi } = {\left\... | Yes |
Lemma 2.5 The Poisson kernel is a good kernel on \( \mathbb{R} \) as \( y \rightarrow 0 \) . | Proof. Setting \( \xi = 0 \) in the second formula of the lemma shows that \( {\int }_{-\infty }^{\infty }{\mathcal{P}}_{y}\left( x\right) {dx} = 1 \), and clearly \( {\mathcal{P}}_{y}\left( x\right) \geq 0 \), so it remains to check the last property of good kernels. Given a fixed \( \delta > 0 \), we may change varia... | Yes |
Theorem 2.6 Given \( f \in \mathcal{S}\left( \mathbb{R}\right) \), let \( u\left( {x, y}\right) = \left( {f * {\mathcal{P}}_{y}}\right) \left( x\right) \) . Then:\n\n(i) \( u\left( {x, y}\right) \) is \( {C}^{2} \) in \( {\mathbb{R}}_{ + }^{2} \) and \( \bigtriangleup u = 0 \) .\n\n(ii) \( u\left( {x, y}\right) \righta... | Proof. The proofs of parts (i), (ii), and (iii) are similar to the case of the heat equation, and so are left to the reader. Part (iv) is a consequence of two easy estimates whenever \( f \) is of moderate decrease. First, we have\n\n\[ \left| {\left( {f * {\mathcal{P}}_{y}}\right) \left( x\right) }\right| \leq C\left(... | No |
Theorem 2.7 Suppose \( u \) is continuous on the closure of the upper half-plane \( \overline{{\mathbb{R}}_{ + }^{2}} \), satisfies \( \bigtriangleup u = 0 \) for \( \left( {x, y}\right) \in {\mathbb{R}}_{ + }^{2}, u\left( {x,0}\right) = 0 \), and \( u\left( {x, y}\right) \) vanishes at infinity. Then \( u = 0 \) . | To prove Theorem 2.7 we argue by contradiction. Considering separately the real and imaginary parts of \( u \), we may suppose that \( u \) itself is real-valued, and is somewhere strictly positive, say \( u\left( {{x}_{0},{y}_{0}}\right) > 0 \) for some \( {x}_{0} \in \mathbb{R} \) and \( {y}_{0} > 0 \) . We shall see... | Yes |
Lemma 2.8 (Mean-value property) Suppose \( \Omega \) is an open set in \( {\mathbb{R}}^{2} \) and let \( u \) be a function of class \( {C}^{2} \) with \( \bigtriangleup u = 0 \) in \( \Omega \) . If the closure of the disc centered at \( \left( {x, y}\right) \) and of radius \( R \) is contained in \( \Omega \), then\... | Proof. Let \( U\left( {r,\theta }\right) = u\left( {x + r\cos \theta, y + r\sin \theta }\right) \) . Expressing the Laplacian in polar coordinates, the equation \( \bigtriangleup u = 0 \) then implies\n\n\[ 0 = \frac{{\partial }^{2}U}{\partial {\theta }^{2}} + r\frac{\partial }{\partial r}\left( {r\frac{\partial U}{\pa... | Yes |
Theorem 3.1 (Poisson summation formula) If \( f \in \mathcal{S}\left( \mathbb{R}\right) \), then\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }f\left( {x + n}\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }\widehat{f}\left( n\right) {e}^{2\pi inx}. \]\n\nIn particular, setting \( x = 0 \) we have\n\... | Proof. To check the first formula it suffices, by Theorem 2.1 in Chapter 2, to show that both sides (which are continuous) have the same Fourier coefficients (viewed as functions on the circle). Clearly, the \( {m}^{\text{th }} \) Fourier coefficient of the right-hand side is \( \widehat{f}\left( m\right) \) . For the ... | Yes |
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