proposition 17.26 The lattice harmonics are an orthonormal basis

Rests on

• depends_on definition 17.25 Multiple Fourier series ¶
  • depends_on corollary 17.49 Lattice sums and the reciprocal lattice ¶
    • depends_on definition 17.29 Fourier transform; the treatise convention ¶
    • depends_on theorem 17.47 Poisson summation ¶
      • depends_on definition 17.1 Fourier coefficients and Fourier series ¶
      • depends_on definition 17.29 Fourier transform; the treatise convention ¶ ↺
      • depends_on theorem 17.9 Dirichlet ¶
        • depends_on lemma 17.6 Dirichlet kernel ¶
        • depends_on lemma 17.7 Riemann–Lebesgue ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:15-fourier-integral-transforms@proof-7 ¶
      • proves proof ch:15-fourier-integral-transforms@proof-30 ¶
    • proves proof ch:15-fourier-integral-transforms@proof-31 ¶
  • depends_on definition 17.1 Fourier coefficients and Fourier series ¶ ↺
• depends_on theorem 17.22 Fejér ¶
  • depends_on lemma 17.6 Dirichlet kernel ¶ ↺
  • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    • depends_on proposition 7.22 Sequential characterization ¶
      • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on equation 7.7 eq:ana-limit-left ¶
        • depends_on equation 7.5 eq:ana-limit-right ¶
      • depends_on definition 7.4 Convergence ¶
        • depends_on definition 7.2 Absolute value ¶
      • proves proof ch:05-real-analysis@proof-7 ¶
    • depends_on theorem 7.7 Bolzano–Weierstrass ¶
      • depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
        • depends_on definition A.36 Cut ¶
        • depends_on theorem A.40 Least-upper-bound property ¶
        • proves proof app:A-long-proofs@proof-32 ¶
      • depends_on corollary A.46 Monotone convergence ¶
        • depends_on theorem A.40 Least-upper-bound property ¶ ↺
        • proves proof app:A-long-proofs@proof-33 ¶
      • proves proof ch:05-real-analysis@proof-4 ¶
    • proves proof ch:05-real-analysis@proof-10 ¶
  • proves proof ch:15-fourier-integral-transforms@proof-16 ¶
• depends_on theorem 12.30 Completeness, expansion, Parseval ¶
  • depends_on corollary 12.19 Double complement; the density criterion ¶
    • depends_on proposition 12.17 The complement is always a closed subspace ¶
      • depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
        • depends_on definition 5.24 Orthogonal vectors ¶
        • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
        • proves proof ch:10-hilbert-spaces@proof-2 ¶
      • depends_on definition 12.16 Orthogonal complement ¶
        • depends_on definition 5.24 Orthogonal vectors ¶ ↺
        • depends_on equation 5.52 eq:lin-orthogonal-complement ¶
      • proves proof ch:10-hilbert-spaces@proof-8 ¶
    • depends_on theorem 12.18 Projection theorem ¶
      • depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
      • depends_on theorem 12.14 Closest point in a closed convex set ¶
        • depends_on definition 12.13 Convex set ¶
        • depends_on definition 12.2 Hilbert space ¶
        • depends_on proposition 12.6 Parallelogram law and polarization ¶
        • proves proof ch:10-hilbert-spaces@proof-7 ¶
      • proves proof ch:10-hilbert-spaces@proof-9 ¶
    • proves proof ch:10-hilbert-spaces@proof-10 ¶
  • depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
    • depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
      • depends_on definition 5.27 Orthonormal basis ¶
        • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.25 Unit vector ¶
      • depends_on equation 5.49 eq:lin-orthonormality ¶
    • depends_on theorem 12.18 Projection theorem ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-14 ¶
  • depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
    • depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
    • depends_on definition 12.2 Hilbert space ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-15 ¶
  • proves proof ch:10-hilbert-spaces@proof-16 ¶
• proves proof ch:15-fourier-integral-transforms@proof-18 ¶

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