Trigonometric Polynomial, 2
Selecting points at x = 0, π , π /2, π /4, the y-values, and the slopes, are simple combinations of the coefficients and √2/2.
Each coefficient leaves certain points and slopes of the curve unaffected.
How many times may the curve intersect the x-axis, between two red points?
For example, Consider adjacent points ( Q to the right of P).
How many times may the curve intersect the x-axis, between two adjacent points?
CASE: If , then plainly f(x) = 0 somewhere between P and Q.
Can there be more than one intersection? If not, and I intersect:
tangent to P,
tangent to Q,
or Segment PQ
with the x-axis ... which is the better approximation?
And other CASES?
E.g. Let R be the intersection of the tangents at P and Q. If , triangle ΔPQR encloses the arc in some neighborhood of , but not necessarily the whole π/4 interval.
Can I draw a conservative envelope which is still a good approximation? I will need over- and under- estimates. Can I do this by forming quadrilaterals and/or triangles by some arrangement of
1. the given points (at intervals of π/4)
2. the vertical lines and tangents through them
3. intersections of 2?
Perhaps I can make a more informed estimate, by breaking up the function into its component parts...