Problem 6

Let

a_1,\ldots,a_n

be positive numbers and let

0<q<1

. Find real numbers

b_1,\ldots,b_n

such that (1)

a_k<b_k

for every

k

; (2)

q<b_{k+1}/b_k<1/q

for

k=1,\ldots,n-1

; and (3)

\sum_{k=1}^n b_k<\frac{1+q}{1-q}\sum_{k=1}^n a_k

Step 1 of 4: Define the numbers

In plain words

The geometric kernel spreads each input across nearby indices.

b_r=\sum_{i=1}^n q^{|r-i|}a_i

Detailed analysis

For $r=1,\ldots,n$ , define the displayed weighted sum. It is positive and contains the term $a_r$ plus positive terms from every other index.