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Problem 4. Let w be a positive continuous function and let n be a nonnegative integer....

Question:

Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,

Problem 4. Let w be a positive continuous function and let n be a nonnegative integer. Equip P.(R) with the inner product (p,q) = $' p(x)q(x)"(x) dx. You do not need to check that this is an inner product. (a) Prove that P.(R) has an orthonormal basis po..., Pr such that deg pk = k for each k. (b) Show that (Pk, pk) = 0 for each k, where the polynomials pį are from the preceding part. Here pé denotes the derivative of Pk. Hint: Since you don't know what w is, you have no hope of actually computing any integrals-and that's fine, because this problem is doable without writing a single integral sign.

Answers

or equal to n. we that is LED au where problem G Pn (IR) is the space of an polynomials of degree lesser know {Vi} a basis foJ-0 ug V. and uk = UK proj. (OK) for all kxl. Now obserserve that proju; (UK) (uk, u;) we claim that, (ej u;) deg K for all KNow let assume NOW let Then uno Vino uk Now as uk uk degree same nool But has we so, that deg (uk) =k for all ock <no.</p><p>For so> Wk-1 = xk-1} - , PK 3. (Notations has been kept unaltered) Now for any K, P ! has degree (k-1). » Pk & span {n9=1, span {uo

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