Introduction to the Mathematics of Wavelets by Miller

By Miller

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Extra resources for Introduction to the Mathematics of Wavelets

Example text

As the Fourier coefficients. Moreover, ✁ 32 ✁ ✝ ✄ ✓✝ ✟ ✂ ✠ ✓✟ ✂ Corollary 5 Corollary 6 ✴✫✬✫ ✄✭✫✮✫ ✁ for any . ✄ ✫ ✁✘✄ ☎✆✄✡✓ ✆✄✫ ✁ ✔✴✫✬✫ ✄✭✫✮✫ ✁ , Bessel’s inequality. ✄ ✫ ✁✘✄ ✡✓ ✆✄✫ ☎✆✄ ✁ ✔ Theorem 15 Given the ON system ✔ ✄✑✓✏✕ ✁ ✂ ✠ norm if and only if ✓ ✟ ✫ ✪☛✓ ✫ ✎✒✑ . ✔ PROOF: Let ☎ in ✟ . For ✗ ✯ ✄ ✄ ✁ ✓✔ ✟ ✄ ✪☛✓ ✡✄ ✓ ✂ , ✫✬✫ ☎ ✔ ✁ ✔ ✄ ✂ ✁ . ✂ ✠ ✟ , then ✓ ✟ ✄ ✪☛✓ ✡✄ ✓ converges in the ✝ ✂ ✠ ✓ ✟ ✄ ✪☛✓ ✡✄ ✓ converges if and only if ✔ ✔ ✁ ☎✁ ✬✫ ✫ ✁ ✁ ✮✫ ✫ ✓ ✟✂✞ ✄ ✪☛✓ ✄✡✓ ✮✫ ✫ ✁ ✁ ✔ ✞ ✓ ✟✂ ✁ ✄ ✔✕ is Cauchy ✄ ✫ ✪☛✓ ✫ ✁ ✓ Set ✄✣✔ ✓ ✟ ✫ ✪☛✓ ✫ .

Lemma 8 ✞ ✁✄ ✆ ✂✁✆ ✞ ✁ ✁ ✞ ✄ is ✆ ✝ ✘ ✝✞✂ ☎ ✞ ✁✝ ✆ ✝ ✌ ☎✆✪ and let ✝✍ . Then ✌✫✠ ✗✖✪✫✑ ☎ ☎ ☎ ✞✓ ✞ ✟ ✄ ✔ ✁ ✄ ✁ ✄ ✂ ✁ and ✁ ✔ ✟ ✄ ✔ ✔ then the action ✁ ✞✁ ✞✁ ✞ ✓ ✞✁ ✞ ✁ ✁ ✔ ✟ ✄ ✔ ✔ ✆ ✁ ✔ ✟ ✄ ✔✠ ✔ ✁ ✟ ✄ ✔ ✟ ✄ ✁ ✟ If ☎ ✁✆ ✄ ☎ ✑ . ✁ ✞ ✁✝ ✆ . D. -dimensional with basis ☎ ☎✎✍✏✍✏✍ ☎ ☎ ✁ and is ✒ -dimensional with is completely determined by its matrix representation ✡☎✎✍✏✍✏✍ ☎✆✠ ✓ then ✔ ✆ with respect to these two bases: ✗✗✪✯☎ If ✂ basis ✠ is a subspace of ✠ PROOF: Let ✁✬✌✓✄ ✁ ✌ ✌ ☎ ✔✠ ✢✗ ✙☎ ☎ ☎ ☎ ✌ ☎ 34 ✘ ✓ ✆☎✡☛ ☎✎✍✏✍✏✍ ☎ ✖✠ is given by ✔ ✌ ✔ ✆✭✠ ✁ ✞✓ ✟ ✄✪ ✠ ✁ ✩✠ Thus the coefficients ✪ of ✠ are given by matrix notation, one writes this as ✞ ✁ ✁ ✄✄ ..

It is an elegant application of the projection theorem. ✁ ✁ Theorem 17 (Riesz representation theorem) Let ☞ be a bounded linear functional on the Hilbert space ✟ . Then there is a vector ☎ ✝ ✟ such that ☞ ✁✘✄ ✆ ✁✙✄ ☎✆☎ ✆ for all ✄✣✝ ✟ . ✁ PROOF: 37 ✗ ✁ ✁ Let ✩ ✔ ✠ ✝ ✟ ☎ ☞ ✁✘✠✖✆ ✞✏✕ be the null space of ☞ . Then ✩ is a closed linear subspace of ✟ . Indeed if ✠ ✡☎ ✠ ✁ ✝ ✩ and ✌ ☎✆✪ ✝ ✁ we have ☞ ✁✬✌✓✠ ✗ ✪✫✠ ✁ ✆ ✌✎☞ ✁✘✠ ✌✆ ✗ ✪✏☞ ✁✘✠ ✁ ✆ ✞ , so ✌✫✠ ✗ ✪✯✠ ✁ ✝ ✩ . , ☞ ✁✘✠ ✓ ✆ ✞ , with ✠ ✓ ✕ ✠ ✝ ✟ as ✒ ✕ ✑ then ✄ ✁ ✄ ✫ ☞ ✁✘✠ ✆✄✫ ✰ ✁ ✫ ☞ ✁✘✠✖✆ as ✒ ✗ ✕ ✑ ✁ ✁ ✁✙✠ ✆ .

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