By Miller

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**Extra resources for Introduction to the Mathematics of Wavelets**

**Example text**

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 ✄✣✔ ✓ ✟ ✫ ✪☛✓ ✫ . 3)✁ ☎✄ ✔ ☎ ✔ ✕ is Cauchy in Cauchy, if and only if ✓ ✟ ✫ ✪☛✓ ✫ ✎✒✑ . D. , if the subspace ✝ formed by taking all finite linear combinations of elements ✄ of is dense in ✟ .

Indeed for some measured values of ✝ there may ☎ be no solution . ✁ EXAMPLE: Consider the ✁ ✂ ☎ ✁ ✄ ✞ ☛ ✄ system ✄✆☎ ✄ ✄ ✄ ✄ ☛ ✝ ✌ ✌ ✄ ✁✂✁ ✁ ✁ ✄✆☎ ✝ ✂ ✪ ☛ ✄ ✝ ✓ ✁ ✁✄ ✁ then this system has the unique solution ✌ ✆☎ ✌ ✁ . However, if for small but nonzero, then there is no solution! 7) for all vectors ✝ and ✞ matrices . We adopt a least squares approach. Let’s embed our problem into ✞ the inner product spaces ✂ and above. That is is the matrix of the operator ☎✯✂ ✕ , ✝ is☎ the component vector of a given ✠ ✝ (with respect to the ✔ ✠ ✕ basis), and is the component vector of ☎ ✝✔✂ (with respect to the ✔ ☎✖✔ ✕ If ✪ ✁✆✄ ✝ ✝ 40 ☎ ✞ ✁ basis), which is to be computed.

EXAMPLE: Consider the ✁ ✂ ☎ ✁ ✄ ✞ ☛ ✄ system ✄✆☎ ✄ ✄ ✄ ✄ ☛ ✝ ✌ ✌ ✄ ✁✂✁ ✁ ✁ ✄✆☎ ✝ ✂ ✪ ☛ ✄ ✝ ✓ ✁ ✁✄ ✁ then this system has the unique solution ✌ ✆☎ ✌ ✁ . However, if for small but nonzero, then there is no solution! 7) for all vectors ✝ and ✞ matrices . We adopt a least squares approach. Let’s embed our problem into ✞ the inner product spaces ✂ and above. That is is the matrix of the operator ☎✯✂ ✕ , ✝ is☎ the component vector of a given ✠ ✝ (with respect to the ✔ ✠ ✕ basis), and is the component vector of ☎ ✝✔✂ (with respect to the ✔ ☎✖✔ ✕ If ✪ ✁✆✄ ✝ ✝ 40 ☎ ✞ ✁ basis), which is to be computed.